Yes but that's marginal because support is entirely contingent on whether the legislative branch members believe that support won't get them voted out.

The US executive is very different because it's an independent election: it's almost impossible to get rid of a President, and relatively easy to deflect blame.

Australia's round of axing prime ministers had some essential logic to it despite the move being relatively unpopular with the electorate: it wasn't about whether the party would lose power, it was about whether replacing the prime minister would let them retain seats they faced otherwise losing.

It's a mammoth difference when the election for executive power and legislative power are linked and it shows.