It's about roster management.
Jake Burger can play a passable 3B and 1B.
We have like 10 guys on the roster who can play 1st. No room there. Jake Burger is also right handed in a roster full of right handers.
Yoan Moncada is the White Sox starting 3B, and the best 3B on the roster offensively, defensively, singingly, and sexily. Yoan is better than Jake Burger in every facet of the game except fairy tail story. Any argument otherwise is trash both figuratively and on any fact basis.
Leury plays every position except catcher. Same with Harrison. They both even play 3B defensively as good, or probably better than Burger does. They can also contribute at 2nd, SS, and in the OF defensively. Burger does none of those.
Danny Mendick has a gold glove nomination at 2B. They were afraid to let Burger even play 2nd. Mendick can also fill in at SS, and in the OF in a pinch. Again, not Burger. Heck Danny Mendick has put up about 100 higher OPS points or a 112 + number, vs Burgers 87.
Gavin Sheets has shown an ability to stand in the OF. Burger has not. Gavin Sheets can provide left handed power. Jake Burger cannot.
When you make up a bench, you want to make sure that the guys you have on it do somethings that compliment the rest of your roster. If Jake Burger isn't hitting, which he hasn't since the Slider train showed up, he brings no value to the bench.