Republicans want to cut all programs.
You keep saying this, but neither their rhetoric nor their actions bear this out. Take the Ryan plan - it significantly increased government spending. And he's considered pretty conservative. In fact, everything approaching a Republican budget proposal so far has increased overall government spending. If Republican's wanted to cut spending across the board, why hasn't anyone in the party proposed these sort of across the board cuts? Or, hell, any sort of overall cut at all?
Government will not cut spending voluntarily, so we need to do it involuntarily. That is, force the spending cuts that happen if Democrats don't agree to compromise.
And the Republican's have made it quite clear they have no desire to cut spending, only to shift it into their preferred programs. And here's an interesting fact: If you are spending more than you are taxing, and you cut spending and also cut taxes by the same amount, you actually INCREASE the size of the deficit. The fact that you claim there is a disaster coming thanks to the debt, followed by you arguing we should do a thing that makes it worse, makes me believe you either don't understand what you are saying or that you are being facetious. If your goal is ultimately to balance the budget, and you believe this is the highest priority, you would only ever be willing to cut taxes if it would increase government income.
The Republican's have pushed to cut taxes, and don't usually bother to argue any specific tax would would increase this income. Ergo, balancing the budget and fighting the debt is, to put it most politely, not their highest priority. Combined with their stubborn insistence on increasing the deficit at all costs, I'd wager it's not even near the top of the list, except rhetorically. (Which sucks, because I'd be willing to accept spending cuts and consider supporting the Republicans if I thought a reduction in the debt was the ultimate goal. It is quite clear, however, that it is not.)
Which leaves the question of "what is"?
What would be a good idea would be to give tax cuts to businesses that are actively hiring and growing. That would stimulate the economy much better then just inefficiently pouring billions of dollars into it.
Also, this is just outright wrong. We've had multiple multi-page back and forths over this issue multiple times over the course of this thread, with a metric ton of evidence brought into play. And tax cuts (at least at our tax rates) of /any/ sort turn out to be terribly inefficient. Turns out the best way to stimulate the economy is food stamps, for a while bunch of reasons. You have to get to rates much higher than ours before tax begin having a decent return on investment.