The President having veto power over the budget is not the problem, the complete lack of any sort of ability to compromise amongst either party's leadership is.

Without veto power Congress could ram through all manner of pork barrel spending without anyone to check them.

The Senate passed the bill 100-0. The lack of compromise is not the problem. The problem is that Paul Ryan is retiring and he refuses to bring the bill for a revote because there's no consequences at all for him.

With veto power, a president could hold the budget hostage until a demand was met.

Yes and Congress can override his veto. That's how the checks and balances work.