They have made the calculation that Democrats are in power; therefore, they will get blamed for any shutdowns. What’s at stake is whether they get the bigger stimulus that the Democrats progressive wing wanted. As long as there’s another party that can be blamed, and as long as that party actually has full control of two branches. McConnell wants the illusion that the Republicans are not infighting and are all in. He sees the Democrats are infighting, and this is what he wants. One thing Republicans have learned from previous government shutdowns and debt limit brinksmanship is that even if public opinion blames them (and it usually does), they will pay little or no price at the polls. After forcing debt limit crises in 2011 and 2013, the GOP went on to very small loses in 2012 and sizable gains in 2014.
McConnel isn’t playing chess. McConnel’s greatest strength as Majority Leader isn’t his strategic acumen or political insight, it’s that he does not give one shit what anyone besides his caucus members think of him, and he will do anything it takes to protect them and add to their ranks. He is absolutely immune to being shamed or cajoled based on public opinion, the opinion of non-Senate-Republican peers, or how he will be judged by history or the hereafter. He has no problem at all being the asshole, even allowing his “moderate” members to criticize him when it helps them politically - knowing that they’ll still back him for Majority Leader. He is in no way bound by precedent, decorum, or his own previous words. He doesn’t feel pity, or remorse, or fear. He is the Republican Terminator. There’s maybe 4 Republicans who care about continuing to promote “bipartisanship” and trying to find conciliation on significant swathes of public policy, so McConnell simply ignores them. He doesn’t fight with them or antagonize them; he just ignores them.