For any non-Americans who are confused, the Republican party isn’t just our right-of-centre party, it’s also our Anti-Federalist party. It’s not really correct that they (speaking in the case of honest conservatives, not psychopathic malcontents like Ted Cruz) think their job is to keep the government from functioning, it’s more that they view it as their responsibility to keep a central government from imposing its will against the states to override the lawful power of the state and municipal governments. A significant portion of Americans, rightly or wrongly, trust their state and county governments to look out for their best interests over the Federal government, and don’t want Washington DC limiting or overriding the force of their laws.
The incident (or series of incidents) you guys are referring too also aren’t as crazy as they sound. They correctly reasoned that, during a midterm election season (i.e. a congressional election cycle with no presidential election) any perceived failure to govern over the last two years would be blamed on the sitting President’s political party. So it made perfect sense–horrible for the people actually living here, but made sense as a strategy–for them to grind the government to a halt during those years. It fit their party’s national agenda (i.e. keep the Fed out of the States), and it got independent voters to turn on the Democrats when they failed to pass their legislative agenda.
I don’t think it’ll happen now though. An appointment to the Supreme Court is higher stakes than a Congressional class or a Presidential election. It’d be a tremendous risk for them to take on; right now Obama’s party is in the minority. He has to pick someone that conservatives can live with, if he wants an appointment to be made while he’s still in office (and of course he does, it’s a big deal for Presidents to have as many SCOTUS appointments as possible). But if they delay and obstruct the process, they have no real guarantee that the Democrats won’t control the Senate again in 2017, in which case they could either force a Republican president to satisfy them, or just appoint whoever they want if they have the White House. And whoever has the White House 2017 - 2020 is going to get to appoint at least two more SCOTUS Justices, and if that ends up being the Democrats I think the GOP is going to seriously regret not just making a deal now for someone that they can live with, to minimize the long term erosion of their agenda.