Tim always loved Robin, way before becoming him, and probably he never wanted to go further, that's why he kinda kept the name. But, what could have added to the character being Dick's partner again? He became Red Robin and started his search for Bruce, something that couldn't have done being Batman's partner.
Being Robin is great, growing up and changing identity is great too. As you pointed out, in the old Earth 2 Dick never leaved the Robin mantle... that's why we don't have a Tim Drake there.
I get he was totally disappointed when Dick chose Damian, but, reading Reb Robin, it's clear that in the end he understood why he did that. He is smart enough, and enough a hero to understood that Damian needed to be saved. He was Bruce's son, after all. A kid who lost everything because he chose to follow his father path. Which kind of hero are you if you refuse to save someone? He loose the mantle for someone else sake, he went through his own path and showed he was right about Bruce being alive. I really don't get what's wrong with that.