A couple of reasons come to mind. I am not saying this happened, but it is driven by the fact that his health has rapidly deteriorated since Alaska.
1) Trump has been at least making a show of trying to stop Russia from continuing to invade Ukraine. Putin wants Ukraine, period. He is not interested in peace.
2) It would be the ultimate power move on the world stage for Putin to kill an American president, especially now that relations are tightening with Russia, China and India.
3) For sport. Putin is notorious for taking out his enemies. While Trump doesn't qualify per se, maybe he is just done with finding him useful for anything.
Well yes, Putin (if it was the real one in Alaska) for sure had the chance to do so in Alaska. But it would also show that the Secret Service really deteriorated in terms of safety procedures.
But alone that Trump took Putin into the beast must have been a constant nightmare for the secret service for weeks. It gave Putin more than enough opportunity to compromise that car and wiretap it or whatever.
We also do know that the Russians have a knack for killing their victims with unusual poisons, like Alexander Litvinenko in 2006 with Polonium-210 or the Skripals assassination attempt with Novichok in 2018.
So it might be that Putin will gain a lot by killing Trump. But since he's also a strategist the other question is which risks does killing the active POTUS involve, and to the gains outweigh the risks?
I am pretty sure that if Trump would be the victim of a Russian assassination that the US intelligency community would know soon enough. And that a response to this act will not take for too long, which could include returning that favor to Russia.
In other words the risks are more dangerous than the gains here IMHO. Killing Trump as Russia would be a real bad idea. This is why I don't believe in that theory, the Russians are not that dumb.
Also Putin got in Alaska exactly what he wanted to get: America put Russia on equal footing by inviting him on American soil. These were really great pictures for Russian propaganda, because a) it added to the Russian narrative that the Ukraine war is in reality a proxy war of America with Russia, where Ukraine has nothing to say and b) that Russia still is a global power rivaling Americas.