Ok, he has to submit a budget request, not necessarily a budget. Such a request doesn't have to be an actual budget though.
I'm not sure where you think you're going with this argument about the budget? The President isn't the King of congress, he doesn't draft their legislation for them.
Even with budget requests, the request doesn't have to be an actual budget. Ergo he still doesn't have to draft their legislation. When he formulates an actual budget they can consider it, and a meraber of congress can introduce it.
This idea that the President has to put in writing a bill to reduce spending....is nothing but political theater. Congress drafts, writes, votes on, and modifies all of these bills. The President can make requests/suggestions, he is required to make a request for the budget. The budget isn't what we're debating right now, the debt ceiling is. He has no role here.