But they don't actually claim the US needs austerity, and they don't push for it either. Republicans are 100% in favor of generous social programs and safety nets so long as the recipient isn't
black
ethnic
poor.
It's not at all about trimming back these programs. The OASI and DI trust funds are profit-positive, earning 4.4% (plus increases due to contributions) on $4.8 trillion in assets. The government doesn't have to "pay" anything to fund the program, because it can already fund itself. The Republican hate for Social Security comes from the fact that, like other large funds, the Social Security trusts store most of their wealth in treasury bonds. Republicans want to destroy Social Security so they can renege on nearly $5 trillion owed directly to the American public in order to honor their obligations to private funds a.k.a. rich folks (on mean that's ~$17k in actual money, a line item from your paycheck, which the Republicans want to take away from you).
I think they're very up-front about this goal.
Edit: To summarize, Republicans don't argue for austerity, they argue for a straight-up transfer payment from the working class of America to the richest 1000 people.