What is the point of 'Soldier of the Quarter' boards?

It's a competition and you start to see the real prestige that comes from winning boards when you hit the Division levels and above. That's when the competition becomes fierce, the Soldiers you're competing against are the best their brigades have to offer and they are actually practicing. Winning boards comes with rewards. The Division board is held at the one or two star general level and being personally recognized by the general of your Division, Corps, or MACOM comes with some great promotion potential. When that General meets you, the Best Soldier in the Entire Division, and asks you what your future plans are, you have the power of a division staff and the recommendation of a General to assist you. When they ask you what your future goals are and you say, Airborne, Ranger, OCS, ROTC, WOCS, Aviator, PCS etc, that General will look at their staff and say, "Make that happen". Next thing you know, you have your Brigade Commander asking if you need help putting your Green to Gold packet together, or whatever it is you want to do.
Plus, you'll learn a whole lot about Army knowledge in the process.

It is a way to set yourself apart/above your peers. It is a test of your skills and abilities. It is bragging rights, not just for you, but for your unit.
Why did you climb the mountain? Because it was there.

They help for overall career enhancement. You will have a record that you excelled against your peers. It quantifies that you went above and beyond just meeting the baseline standards. I don't know how the Army does it. The Air Force, typically at your level will go to an Airman who performs their job at a proficiency level well above their peers. They are the ones who can do their job without being instructed. They are the ones who are also doing things for self improvement like getting their associate degree though the CCAF and are typically going further to get their Bachelors. They are doing volunteer stuff like participating on the base honor guard for example. They are the ones who aggressively working toward their journeyman 5 level in their AFSC. I would say like in a squadron of 30 Airman. 20 will be average to above average. A small handful will struggle they might be able to do their job well, but don't take initiative. Of course there is the rare dirt bag. Then there will be like three or four who are well above average to their peers. They are the ones who show that they are capable to move further in their career. Those are the ones who get those awards. In the USAF if an E-4 wins the wing level, they are typically step promoted to E-5. Those are the ones who if they put in a packet for a commissioning like a full ROTC scholarship or in some cases an appointment to the academy, they get it. Others who do not desire to go commissioned are typically getting groomed for upper level NCO leadership, like eventually to possibly be a Chief.

Been there, done that... (Former company and battalion Solider of the Month back in the day.)
I still remember the only command that doesn't require a Soldier to take another step, "GAS!".
Where is the only "truck" on an Army post? The top of the flag pole... The follow up question was what does it contain? A razor, a match and a bullet.
See where I'm going? Most likely you're going to say no.
It may seem like useless information at the time, but what we learn bonds us with those who have gone before and we can then share with those who follow on in our footsteps.
The line is long and I pray that it is never broken.
You can leverage the knowledge you've gained from the Soldier of the Month, Quarter, Year Boards at the Company, Battalion, Brigade, Division, etc. levels and try your hand The Sergeant Audie Murphy and Sergeant Morales Clubs.
The knowledge gained will serve you well throughout your career and even after you leave the service. Hence my wealth of useless information I share on this forum from time to time. What good is it if you don't choose to share it? Remember that your computer may fail and Google won't be available to answer your questions... Once you know something, hopefully you won't forget.

SSG (Join to see) SFC O’Mally’s response is correct. Climb the mountain! Enjoy the view!

The route sometimes becomes arduous, that is the enjoyable part, are you really a superior Soldier?

BLUF: The confidence to LEAD Soldiers!

This is a great question, and I have enjoyed reading all of the responses. I will honestly admit that until about about a year ago I was not a huge fan of boards as well. My tune has changed dramatically now that I’ve seen what this process has particularly done for one of my Soldiers.
I’ve been at my current unit for 11 months, and I’ve seen this Soldier grow exponentially during this time. I saw the potential he possessed, and we discussed what he should do to take the next step in his life and career. I could see that he lacked confidence in his own abilities. He is a hard worker, pretty good technician, above average fitness, very observant, and he is competitive. I felt he possessed everything needed to be a good leader, he just needed to believe in himself.
He went to the promotion board today and ROCKED! This is a Soldier who, 6 months ago, could not recite the Soldiers creed aloud because of nerves. We put him through several mock boards, and a couple of BN SOM boards. He didn’t win, but it gave him the push and confidence he needed.
It also helped his first line supervisor, which I am grooming to take the next step in his career as well. He was on 6 month school option, but all I heard was how good he was. And to his credit he is definitely one of the best 25B technicians I’ve encountered, and Also very honest. He tells me that he could run a help desk with ease, but is lacking on his military know how. TOO EASY! Lets move out and draw fire. First things first, stop doing all of the tasks I give you, and provide our two junior enlisted Soldiers all of the tools so they can be technically proficient. Secondly, do everything in your power to develop them to take your place because I am definitely doing that for you. Through this process of counseling, development, and preparation both Soldiers have attained KSA they didn’t have before.
Yes there are an insurmountable amount of accolades that come from winning, and we as leaders have to do out due diligence not to produce “board babies”. But I’ve learned that the board process means so much more.

I will preface my response by saying that my experience with boards is limited. I have only recently just competed in a board. I competed in both the BN and the follow-on ASG-Qatar NCO of the Quarter boards. I won both. I had never competed in a board prior. The opportunity to compete in boards is much rarer as a member of the National Guard. I was encouraged by my squad leader peer and my PSG, both of which were former-active duty.
It turned out to be a great experience. It was a lot of studying and memorization, some of which was knowledge that I will likely never use. However, it absolutely made me learn things that I might not have otherwise known about. It broadened by overall knowledge and gave me the opportunity to showcase myself.
Beyond the knowledge I gained, it helped earned bragging rights for my squad and my platoon and helped to set us apart from others in our BN/Task Force. Esprit de corps is very important to me. It gave us something to be proud of as a group. But it also set an example for others. Immediately after winning the BN board, I had a team leader and a Joe tell me that they wanted to study and compete in the next Soldier/NCO of the Quarter board.
I studied with a Soldier from another squad (same platoon). He did not have an NCO mentor available (due to our mission) and so it gave me the opportunity to help a Soldier I might not have otherwise interacted with closely. He also won both boards and we both agreed that was a worthwhile experience.

It’s existing in a different space than the rest brotha. Always be the water walker amongst your peers.

Real quick answer to your "point(s) behind the Board" is "all of the above".

Reason being, the services love a "Complete Soldier", who is one that's not just competent "in their field", but one that's also well versed in Military / Service traditions and histories, regulations, current events, etc., and one that's able to function in "stressful" environments - at least as stressful as they can generate in Garrison...

Though to be "fair", things like "recognition" and "promotability" are "side effects" of the process, the "heart" of it is supposed to be the "friendly competition" to prove who "the most complete soldier in the _____" is...

I've never been a fan. They test very general skills, and who can study archaic stuff the best. Lots of people who stuck at their MOS but are good at studying and "appearing" great. If it doesn't take away from the mission or make crop soldiers appear great, sure why not. But when Soldiers are forced to go or the bad ones win. Then there's a problem.