Well, it does matter, because the White House’s political strategy rests on blaming the Republicans for it.
As far as fixing the problem, it is the blame game, and the public’s reaction that will determine which side has to give in. If the public thinks the Republicans are being obstructionist and stopping a reasonable compromise solution, then the Republicans will have to cave and raise taxes again, or else lose the next election even worse than the last one.
If on the other hand, the public decides the White House is a bunch of liars who constantly move the goalposts to fit their whims, then it’s the Democrats who will surrender or face the wrath of the voters.
So you’re wrong, the blame game is the only game for the time being.
Congratulations, you should be in congress. You have bought into the talking points.