Why does Army BCT not include Crew Served Weapons in its POI?

Had to be a time and funding issue as others mentioned.
I graduated BCT at Jackson and we did the 240, 249, M2, and MK19.

I agree with everything the group has posted and the additional decision by each service component is to rely on the individual units to complete these basic military tasks. This is where the leadership of each small unit has the requirement to complete Fiscal/Annual Training Events. In the event a new join to a unit isn't certified it is on the unit to complete these events per member - this goes into the unit readiness cycle (the deployment decisions come from these statistics.) Those of us in the training arena do the best we can to get our units to a close to 100% as possible, this keeps the boss out of trouble. As a Marine every other year a unit has to complete a Marine Corps Combat Readiness Evaluation (MCCRE), every individual will be evaluated and graded. The unit commander will have the overall grade included on their annual evaluation. A similar process is used for each unit functional area (Administration, Sustainment, etc...). This is why we say training never stops, think of Olympians (their objective is to get the gold). We just want to get to the battlefield, complete the mission and get back home safely. "The more we sweat in training the less we bleed in combat."

If your son wanted to fire more weapons have him reclass to 11B.

Your question is valid but the challenge is that the answer isn't very simple. There was a time that training to be a leader didn't come from any formal school. It came from NCO's and Officers that had "figured things out" and developed a knack for it then subsequently passed it on to their subordinates. This included training on anything to any extent but at some point the military fell into what i have come to call their "age of Aquarius", a term taken from the 70's hippie movement, where they decided that wasn't sufficient and they needed a formal school to develop leaders (instead of through their officers and NCO's) and it has gained traction from there adding more and more to the saturation point we see it at now.

The addition and/or subtraction of specific training in BCT seems to follow this same pattern of logic. They seem to imply that the trust in officers and NCO's at the unit level to train Soldiers adequately in certain disciplines, functions or equipment is so low that they feel they need to do it themselves or conduct it at a time, place and manner that they can be sure the training took place and to standard.

Now I can't be entirely sure but that's my take on it.

Chances of FNG getting a pig are slim to none. He would be ammo bearer and assistant gunner before he was gunner. Plenty of time to learn.

Back In 79 At Marine Basic, We Qualified With M-16s , Threw Grenades And Fam Fired 1911s .
When I Got To My Permanent Duty Station As Huey Aircrew We Were Issued S&W 45 Revolvers And Trained On M-60s , M-2s And GAU-2B Chain Guns

When I went through BCT/AIT in the mid 60s we trained on EVERYTHING an 11B would use (I was an 11B) from the M14 to land mines, everything in the Infantryman's catalog before we went to AIT, iirc, it was only about an hour introductory session on each weapon. Some in person, like hand held/crew served weapons, others like landmines were in classrooms.
In AIT we got Advance hands on training on those weapons (inc. one day on Mortars). Later I switched to an 11E (Armor crewman), but I still retained those skills. How much of it I would ever use again would be questionable (I never did), but I had a basic understanding of how to use it should the need arise. To send a non combat arms soldier into an area where trouble may happen suddenly, without warning, as has happened, without the basic skills to operate a light/heavy machine gun or Mk 19 is the same as sending them in unarmed, it's unconscionable.

It comes down to time. BCT schedules are VERY strict and are set by TRADOC; unfortunately, there isn’t a lot of time dedicated to familiarization of crew serves. Funding is also a huge part; it costs roughly 80k to train one soldier, not including the cost of ammunition, and trainees run through 7.62 like cough drops. Detailed crew serve training just isn’t sustainable.

We were trained on 50 cals and m60s when I went through basic.

They took "US Weapons" out of the BCT POI about 3 years ago, not exactly sure why, probably money.