"Trump's coziness" is nonsense, a result of propaganda from the Democrats. Trump took actions that hurt Russia, especially by undercutting their economy through encouraging hydrocarbon production.
Trump is a deal maker, and deal makers tend to make kind comments about their opposite number, even if in private they think they are snakes. Trump's comments get picked up by the biased left wing media, aided by the bogus "Russia collusion" investigation (which didn't find any), and used as propaganda against Trump.
Don't get me wrong, I have issues with Trump. But cozying up to Putin - it just isn't so.