I always wondered why he suddenly becomes allergic to citrus.

In one of his first appearances on SG1, he invites Carter to the cafeteria, mentioning that they have lemon chicken, with excitement. If I'm not mistaken, it's not until the SGA fork that he seems to be allergic to citrus.

I think you're confusing episodes. In his first appearance he mentions that he's allergic to citrus. The "I love lemon chicken" incident comes from Carter's visit to a parallel universe when she's experimenting with the merlin device in a later season.

...I'm really embarrassed that I know that from memory.

He might've developed a sensitivity to it in the meantime; that sort of thing happens.

How do we not have a video game franchise about at least one of these three great settings?

Probably the same reason we don't have more content from them.

My own supposition is that it's the standard rights-squatting thing a studio does when they're done producing a given work and don't want to compete against 'their' IP being used by someone that actually has a good story to tell.

While entirely possible, it's a weird stance to take. More exposure on different/new mediums would lead to more/renewed interest in the media on other mediums, in my opinion. A cool new Stargate game might inspire me to go back and watch old stargate (or finally see some of the movies I missed).

There were some: https://en.wikipedia.org/wiki/Stargate_games#Video_games

All of them failed. I really don't know why MGM didn't gave it to a AAA studio, such shame...