if Ukrainian nationalism has been hardened due to the strikes within Ukraine, then the same can be said for the Russian people

Arguably there is a difference. Ukraine had NATO-funded propaganda machine pissing in their ears with ethnonationalist propaganda for over twenty years (at least). On the other hand, Russia is multiethnic, which causes certain gripes to the ruling class, because they are forced to tone down ethnonationalist rhetoric at least sometimes. People from Russia, it seems, are more likely to reject their professed nationality - from various local nationalist movements to "I am ashamed to be Russian" campaign launched immediately after SMO began in 2022.

That said

but there is good reason for concern that they will amplify his claims that Russia is fighting NATO, not the Ukrainian people.

I am morbidly amused that the media is still trying to maintain this particular farce

Yeah, it's a nonsensical idea. If simply doing deep strikes could win the war then Russia would've won a long time ago.

It's worth noting that Putin is a moderate in Russia, and if there's any criticism of him domestically it's that he's being too timid in Ukraine and dealing with the west. If anything happened to Putin, it's guaranteed that somebody like Medvedev, who's far more hardline, would be in charge instead.

I'm pretty sure killing the Russian president (and likely his cabinet, which would always be nearby) is plenty of justification for nuclear war.

I'm sure if Russia bombed the White House and the president, nukes wound start flying immediately.

Well depends on which war. They could hit objects that will hinder Russian ability in future conflicts with other NATO lapdogs - see for example strikes on nuclear warning radars. Plus there's a history of successful drone attacks on Russian airfields (because for some reason jets are stored in the open there). Those jets are used for retaliatory strikes