Congress could pass a law that every bill will deal with one issue and one issue only, but this is quite unlikely to happen.
The majority party can make this a House or Senate rule, and then it won't even require consent by the other party.
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They could, but they won't. They need the omnibus type bills to pass the sort of things they can trade to other members for a compromise.
Nevada needs laws that favor mining, Nebraska needs laws that favors winter wheat, Rhode Island needs laws that favor fishing and ferries. No one in Nevada cares about wheat or ferries. With omnibus bills, they can agree on favors for all.
This was the problem that Jimmy Carter was faced with. Congress passes costly laws that favor only a few, and when it does not, it drags to a halt. Carter wanted to simplify government and make it more open and honest, and Congress resisted all his efforts to do this, even when he was scoring 70% in the popularity polls. Eventually, they ruined him. There are clearly limits to what any president can do. It's always far less than the expect they can do before they are elected.
Our government is deliberately designed to resist change. Parliamentary democracy makes it much easier to change.
