> Genuine question: how much does Russian public opinion matter

If it didn't matter, internal Russian propaganda organs wouldn't be firing on all cylinders. The press is not just a 'jobs' program.

Every government requires the consent of the governed. Democracies, semi-democracies, dictatorships, autocracies, theocracies, all of them only work because people believe in them.

For this reason, all forms of government put a lot of effort into manufacturing consent among the people they govern.