Clearly you haven't been following this in detail.
The reason the Republicans need a supermajority is that they have not been able to keep enough of their own party together to get a regular appropriations bill passed. So they turned to a stop-gap parliamentary procedure called a "continuing resolution" in which both sides (a supermajority) agree to allow spending to continue while they work out their differences.
So this is all about making a deal with Democrats because they can't keep their own party together. In your system that would trigger an election.
Now as for immigration, a deal on the DACA program, which literally nine out of ten Americans favor, would bring enough Democrats over to pass the continuing resolution. But the Republicans added another condition to that: funding for the President's border wall, which if you recall Mexico was supposed to pay for.
So the Republican leadership has reached an impasse in which they can neither muster enough votes from their own party to pass a regular appropriations bill, nor enough votes from the opposition party to pass a continuing resolution.