In general, why is modern society so quick to cry "racism"?

Racism is an issue, but then again, it has been an issue, in all societies, throughout history. It has taken different levels and forms, but in the history I've studied one group of people has looked down on another group of people for a myriad of reasons throughout history.
I don't think racism is more prevalent in todays society, at least in the U.S., than say when I was a kid in the 60's. We've come a long way, but as mankind is still in the evolutionary process, we've miles yet to travel.
Why is it a rally cry?? I suppose it depends on who is doing yelling. I didn't get the job, cause I'm black, or I didn't get the job because I'm white. Maybe neither got the job because they just didn't meet the employers criteria. But instead of, Know yourself and seek self-improvement, it's easier to lay blame on the idea that racism was the factor.
It's an interesting question.

Because when the left has no more legitimate arguments or facts they claim racism, sexism, mysoginism, narccisist, homophobe or whatever other moniker sticks cause that is what people remember. Sad state of affairs.

Because it's popular and quickly gains press attention which brings pressure against those trying to honestly deal with real issues.

Why? Because it is easy!

We live in a society that loves quick easy answers -- and tagging someone or something with a label makes it easy to either deal with it or ignore it.

Racist, Nazi, Socialist, Liberal, Right-winger, *-phobe, are all easy ways to dismiss someone as being beneath any consideration. None of these labels require any proof, none of these labels require any thought beyond "I disagree with you"

It's a "convenient," way to do a couple of things. One, it makes the person being accused stop and think about their actions/conduct which may or may not be valid. Two, it taints the person being accused of racism, whether true or not. In short, it's the blame game and may or may not be legitimate.
For false accusations, there should be some cost to the accuser and that rarely, if ever, happens. For true accusations, there is probably always a cost to the racist actions/conduct, at least to their reputation and thus, credability as to future events.
Today, offending someone is all too easy. It seems people (regardless of race) are EXPECTING, or anticipating being offended; maybe even HOPING to be offended. It's insanity. It's unhealthy and it's extremely divisive.

It's the quickest way to stop the conversation. Snowflakes don't want to debate because they are afraid of anyone who holds an opinion counter to their own.

Short answer: It works.

Long answer: It has become all too easy to blame someone else when you are in the wrong (or equally wrong as the other person) under the guise of "they hurt my feelings," or "they were discriminating against me." Political Correctness is a poison that has seeped into our society and is destroying us from the inside out, just as sure as rattlesnake venom. And due to everyone's death grip on PC mannerisms, race and gender are the two hot topics that have worked up, down and sideways because they have historic precedent. In our Nation's past, both race and gender were the source of the biggest forms of oppression and discrimination. And because of this historic precedent, it is easy to claim such discrimination with a "old dogs/new tricks" mentality. Because of the fact that it has worked so well in the past, it has become the go-to, because people now know that their cries will be met with sympathy (either public or judicial).

I imagine that for those who have never experienced upfront in your face racism where your life was possibly on the line, it is hard for them to even emphasize with those of us who have. For instance, at 12 years old while walking down Old Galveston Road in Houston TX some hooligans in a pickup truck threw a pipe which hit me right above my right eye and then jumped out yelling "GET THE NIGGER!". Luckily I knew the area and was faster than them and even with blood streaming down my face, I was able to outrun and avoid them. And NO, that experience did not deter me from being who I am now in my 70 plus years on this Ball of Mud, and accomplishing what I needed to. So all that left of that event is the memory and a quarter inch scar over my right eye. As a matter of fact I don't think of it unless some commentary such as this shows up which tries to discredit the experiences of people who suffered from what some try to say doesn't exist.

The most egregious, heartbreaking and sad memory of mind is from 1969 after my first tour in Nam and was on Leave with one of the guys from my unit who lived in the same area of me. We were to meet up and he and his brother and a couple of friends for a few beers. The place they decided on was a "GO GO BAR". Everyone was white except me and that changed the entire dynamic of the evening. Even though I was decked out in my Sumner Khakis with all the salad on the chest, the bartender still stated; "HE CAN'T BE IN HERE." On the real that didn't even matter to me because I was born in Texas and even as a child noticed the hatred. Didn't understand it but it was hard to miss. My white brethren took more of an offense than I did and a fight broke out. Luckily one of the cops that showed up had served and was sympathetic which kept us from going to jail.

The above didn't change the trajectory of my life after experiencing this because I was and am a student of history, especially military history. I had the privilege to speak with some of the original members of the Montford Point Marines and listened to what they had to go through simply to be able to serve their country. I read on the WW II black veterans who came home from battle to be denied the same benefits as their white counterparts. While the GI Bill’s language did not specifically exclude African-American veterans from its benefits, it was structured in a way that ultimately shut doors for the 1.2 million Black veterans who had bravely served their country during World War II, in segregated ranks. When lawmakers began drafting the GI Bill in 1944, some Southern Democrats feared that returning Black veterans would use public sympathy for veterans to advocate against Jim Crow laws. To make sure the GI Bill largely benefited white people, the southern Democrats drew on tactics they had previously used to ensure that the New Deal helped as few Black people as possible. So wouldn't we consider that as being systematic of "racism"?

I see some saying "That was then. Why can't you get over it!? Makes me wonder what trauma they had in their lives that they got over so easily? In the years since, while always working in a professional field, I have a few times been stopped while driving and/or walking black. Back in the day spoken to disrespectfully and detained because "You fit the description!" But that didn't stop anything either because you learn to play the game and subsequently teach your sons and daughters how to play the game so that winning means that you get to make it home while many others did not.

As a student of history something that is happening today really brings it home. Children like Ruby Bridges had to carry the memory of being escorted into school while angry mobs called her names and threw stuff at her and she is still alive and prospered despite of how she was treated. But the fascinating aspect of this is that even though a young black girl had to endure this treatment, some are saying now that her story is not worth being told and they are trying to strip it and other atrocities from the annals of history because they say "It makes little white boys and girls feel bad about themselves" apparently paying no attention to how it made all those young black boys and girls feel. So that is not considered "racism"?

We won't even delve into the legal system where two individuals of similar backgrounds, even when one of the defendants are charged for the first time and the other has multiple cases and even convictions gets lesser time than the individual with no priors. It is all recorded And that is what is called "Systemic Racism". Of course, you yourself might not be racist or have preconceived racist notions, but to deny that it does exist is in and of itself perpetuating the plight on humankind.

That is my soapbox performance for now and I'm sure that some won't understand and won't even bother to try to understand and that is alright. Look at the world and its people. Look at what humans do to one another, some even in the name of God, so at this late date in the game, I have no expectations for people because that always leads to frustrations. And trying to deny history does not erase it. It is what it is. I totally agree that everything is not racist but it is undeniable that some things are. Everyone is not racist, but there are racists among us. To paraphrase the radio serial The Shadow; "WHO KNOWS WHAT EVIL LURKS IN THE HEARTS OF MEN?" They answered that "THE SHADOW KNOWS! But I will change that to "YOU KNOW WHAT EVIL LURKS WITHIN YOU" And should not "TO THINE OWN SELF BE TRUE"?

Based on how you even termed your question seems completely dismissive in the first place. Perhaps people, as you so dismissively put, "cry" about "racism", is actually someone being racist. It's best to dissect the issue before forming an opinion. I like to call it out as I see it, whether it be a passive aggressive or blatant remark from someone.

Because 'modern society' cannot argue using facts and logic. So, if they can't disprove your point, or prove theirs, the only other option is to gaslight you, demean you, or somehow besmirch your character. I cannot count the times I have been accused of racism, fascism, misogyny, white supremacy, hate, anti-semitism, or any number of other pejoratives when I have argued with facts and logic and my 'opponent' cannot do the same. It's vulgar, disgusting, and lazy.
The saddest part to me is that my generation worked so hard to rid America of those horrific 'isms', and now I and my generation are being accused of those things. It's disgusting and false.