Then it is the fault of the Democrat Senate and president, but the House has performed its Constitutional duties.
You seem to have forgotten that your original point was: "It was never the Founders’ idea that a majority in the House only act if a Senate majority and the president go along with their view on things." These two points of yours are precisely contradictory.
The founders did, indeed permit the Senate and President to concur on all bills, including the budget. Also, the House did pass several partial funding bills, and the Senate refused to even vote on them. I still say, what next? You now have the government shut down, in your scenario, and you're doing nothing.