cut funding off at the knees by vetoing any and all budgets that contain it.
Nah. He'd have signed a continuing resolution, which he did multiple times during his Presidency when he couldn't agree with Congress on cuts. And don't forget that he wanted to repeal Medicare and Medicaid, but ended up dropping it because there wasn't enough public or Congressional support to do it.
There's only so much a President can do if he doesn't have a working majority in both houses of Congress.