Is there a second best option you would suggest? Just don’t put a cabinet on the counter?

I have a cabinet that goes all the way to the wall. Leaning over the countertop to open it is no big deal. BUT it's a 22" cabinet in the corner, with 2 doors. Opening a door towards me, then reaching around to get the contents, is unpleasant. If your corner cabinet is around 24" or less, have one door that opens away from you (toward the wall).

Post your plan here. Based on your description I would say skip it.

Sounds pretty dysfunctional. Who is designing this? Not being a cabinet pusher. Designing the whole?

I don't like appliance garages in general. The doors take up a lot of interior space of the garage. The garage doors are an expensive upcharge. You lose valuable counterspace with the garage. You have to add to the budget the electrician installing an electrical outlet inside of them. And as mentioned, people tend to leave the garage doors open in any case. But setting all those reasons aside, just from reading your brief description in your OP, your kitchen design sounds problematic. Might have to start over.