Re: I dunno, he kinda reminds me of...
There's at least some coherence in applying the 'game' thing to Jason, which was already popped out of a hat used in Death in the Family - which really should be a warning all on its own to take it with a pinch of salt, but at least there's some previous canon basis for it.
Bruce thinking Dick was Robin for the 'thrills' of it... not so much.
Tim is the best Robin in some objective way
I think Tim is the only kind of Robin Bruce was likely to take so soon after Jason's death. More cautious, less impulsive, rich, and with parents. Bruce had adopted Jason, and Tim having at least a father alive meant that this time the roles wouldn't get confused. In the long term, it didn't work out that way, obviously, but at the time it must have been a reassuring difference for Bruce.
Also I think that the very things that make Tim efficient and reassuring to Bruce are precisely the ones that make him the least Robin-like of the Robins.