I have a very simple integer linear program and it is really waiting for the heat death of the universe.
No, running it as a linear program is still slow.
I'm talking about small n=50 taking tens of minutes for a trivial linear program. Obviously the actual linear program is much bigger and scales quadratically in size, but still. N=50 is nothing.
Yes, there are also instances of problems in NP that are hard to solve in practice.
But here again: solutions to your problem are easy to verify, so it might be interesting to let an AI have a go at solving it.