Putin also probably doesn't want to burn every bridge with Israel if he can still avoid it. he seemed to very much appreciate the substantial volume of Russian speaking, ex-ussr Israeli's as something of geopolitical worth (and a point of connective pride within his nationalistic outlook), he had a good relationship with Netanyahu in particular, even despite the issues caused by opting to get involved in Syria for Assad. things had only recently turned sour because he's backed himself into a corner with the invasion of Ukraine. Israel didn't care about Crimean annexation and were even relatively slow to fully condemn Russia immediately after the invasion until it became apparent what a military trainwreck it was (and likely significant pressure from the states/ europe).
No Ukraine war and i'm pretty sure he'd be firmly on the Israeli side post-Oct 7th, though likely with some self-serving, fake attempts to appear more humane and eager for ceasefire than the Americans/Germans/Brits.
Good points. I recall there was significant coordination between Israel and Russia during the Syria war, where Netanyahu flew to Moscow to deconflict Israeli airstrikes on Iranian proxy targets with the presence of Russian troops in Syria at the time.