I don't think Trump is explicitly an asset. He just likes oligarchs and dictatorial strongmen, and is usually the dumbest man in the room. There's no doubt that there are ties there, but Trump is not a loyal man. These things combined means he can swing back and forth between doing Putin favors and being genuinely upset by perceived slights, then back to friends when Putin gets his ear to smooth things over.

Rather than specifically being a Russian asset, he's an asset to the last charismatic man he remembers speaking to.