The whole idea of a site not auto time banking seems strange to me. When do you need the time bank? When you run out of time. So why not just go into the bank when the time runs out. Seems so odd.
But your timebank is an (almost) one-time resource. When you are involved in a big pot in another hand causing you to timerfold pre somewhere else you probably have junk anyway, and the EV difference of folding say JTs in the CO instead of raising ain't enormous. There's a much larger EV difference for timerfolding to a 3b/cbet/riverbet or whatever once you've already put in $ though, so it makes sense to only auto-activate the timebank once there's money from your stack in the pot so your timebank doesn't get wasted on the situations where using it has the lowest value for the player (and in fact is probably -EV to use since you can't use it in much more important spots later anymore).