Yes, but assuming the amendment were limited to the budget, that's a good thing.
I disagree. What you're proposing is basically saying it's OK for Congress to keep adding pork to bills, and it's the President's job to be their babysitter since they are incapable of acting like adults. Absolving Congress of responsibility for their own actions and placing it in the Executive's hands isn't a solution, it's barely a band-aid.
The better option is to force Congress to take responsibility and have the bills come out clean in the first place. The obvious solution is to change the internal procedure that allow sunrelated amendments to legislation. The less obvious but more likely (in a pig's eye, I know) are to stop making seats artificially safe for party boss and big donor favorites through gerrymandering and to rein in the money.
I focus on what's practical.
How is adding yet another layer practical for anyone but Congressional party leadership?