Would they really have allowed Forrest into the military?

From what I understand the draft made everyone have to join and they weren't as strict as they are today.

God Bless the people who have served in our military, but based on some of the ones I have known, the military doesn't require a high IQ for enlisted soldiers.

And as we saw in the movie, Forrest seemed to do very well in basic training, and ended up winning a medal of honor for his only experience in combat. Perhaps not being very smart helped him, in that he didn't let them mess with his head, and he could just focus on the task in front of him.

He was a college graduate who could follow instructions. I'm in the Navy at an Officer training command and Gump seems more qualified than at least half the ensigns I come across.

Plus, I gotta think if being simple minded were a way to avoid the draft, a lot more people would have tried it back then.

They always need guys to sweep the barracks or swab the deck. In spite of what recruitment commercials would have you believe, not everybody is sitting at a computer operating drones.

Whatever the case was he got in. And they were short on their quota.

Watching the movie again recently, even though Forrest is slow-witted to his peers, his actual physical dexterity and mental acuity are better than most people around him. Basic training weeds out the physically and mentally weak, and while the perception is that Forrest may not be mentally astute, his awareness for others probably exceeded his fellow training mates who were probably more self-conscious than he was. There's no room for ego in military training, just straightforward commitment and physical endurance. Forrest has those in Spades.

I don’t think this movie cared enough to consider that

Yes. The IQ requirements for the military was indeed lowered (primarily because many volunteers turned out to be low IQ) and Forrest would have made it in.