Experienced old grouches or not, I feel bad for the capabilities of your local shops. Not only are they incapable of extracting a stuck BB cup they also don’t know how to cut down a fork crown. I do wonder if their hesitation isn’t that it would be cut to the correct size so much as it would be cut to the next smaller standard than what you’re expecting.
While I wouldn’t call it incredibly common, the surfaces on a bike that require a finish ream/face are out of spec. are often enough that I wouldn’t call it uncommon either.
As for which part to modify, I don’t like turning my bikes into art projects by custom fitting parts that should be off the shelf items, so whichever part is non-standard should be corrected. The only exception I’d make is if the fork had been cut incorrectly undersized already I’d consider shimming it back up. Done properly this is fine, but it sounds like your problem is the other way.
It’s much easier to hit the intended tolerance with the made for purpose hand reamer (our local co-op has the Park one) and is a 5 minute job on a bare fork. It’s perfectly doable on a lathe as well, assuming a competent machinist. The only risks there are said machinist being lazy and cutting the race off center, leaving too large a radius at the bottom, or straight up missing in size and trying to call it close enough. None should be an issue if they know what they are doing…but neither should cutting out a stuck BB be an issue for the same person.