Well first, since the t stat is next to the bottom hose, it should be almost the same temp as the top when it fully opens. If it gets warm like you said, it may be sticking upon opening. Which I've had to go through 3 different t stats on one car, before getting one that opens correctly.
I would start there because, if you can make the fans come on manually, there has to be a reason why. And i think because you are getting a partially stuck t stat, that might be why the fans aren't coming on at idle.
And if you just installed the new t stat and nothing else, that would lead me to believe that is the problem.