Eh, NATO only acts like the "puppet club" because most of them don't have their own standing military. I think if Russia joined they would have far more sway than many of the current NATO members since Russia would be a huge boon for the US to work with as an ally.
That being said, tensions with Russia have grown considerably under the previous 2 administrations so there would have to be some serious mending done before any talk of a true alliance could be done. Russia would also have to clean up some of its human rights and free speech issues before they would be granted entrance to NATO, which they may not be willing to do or find worth the effort.
I think the most likely scenario ends up being solid economic/trade deals between the US and Russia under Trump, with some potential for military cooperation down the line.