The omnibus bill was not a budget. It left major holes in the actual budget, just taking care of the immediate emergencies. Those holes will need plugging. There are always new spending bills that also need passing.
Then there is the debt ceiling, which if it is not raised the omnibus bill becomes meaningless. The Republicans always have a minor temper tantrum with that. This looks like a preview of that disaster.
We can obviously survive a week without a House. We can probably survive a month without a House. Survival starts looking less likely as the months progress. I doubt we could survive until October.
And remember, it will be the same Republicans slowing down the process for the next two years. Imagine we get a House in August, have a week to pass a debt ceiling, and two months to pass, negotiate, and repass a budget. That would require very quick work, from a Congress that took 7 month just to swear in (a process that usually takes an hour or two).
I guess we can be grateful that this is not a Presidential election year. If this had happened in 2020, and was not resolved by January 20th, we would not be able to have a President. How long do you think America can go without a President?