The GOP won a crushing victory in the 2010 midterm elections. While this was undoubtedly due more to the slow pace of the recovery than any overriding public concern about spending, the deficit did figure prominently in the campaign, so there was a plausible argument that America was demanding large spending cuts. The debt ceiling crisis of 2011 should be seen in that context.
But what about today? The GOP lost the presidential election in 2020. It lost a Senate seat in 2022. It underperformed, relative to history, and won only a tiny majority in the House. Culture war issues were the focus of the campaign, not spending cuts. Finally, the GOP has shown throughout recent history that it only cares about spending and the deficit when a Democrat is in the White House. The electorate is perfectly aware of that.
It is consequently absurd to suggest that McCarthy has some sort of mandate to impose spending cuts by holding the economy hostage. Given the divisions even within his own caucus, the only mandate he has is to keep the lights on and wait for 2024.