I think Tactics Trainer should be called Tactics Tester.
Recently, I had occasion to contemplate what sort of practice might improve your game. (Actually, the "occasion" was the "high IQ thread. There was a lot of good information in those threads. Pity it was allowed to be highjacked, but that's another story.) At any rate, I concluded that slow, deliberate study was better. Therefore, I went back to TT, and decided I would completely ignore the clock, and just look at problems until I solved them, and was sure that I had solved them. I think that is much better training than to see a clock ticking down and take the best move you have seen so far, even if you haven't analyzed it all the way through.
Today, I went 21 for 25 on Tactics Trainer problems. My rating fell five points.
That in itself is not a huge problem. If I don't care about my rating, then I don't care if it goes down. However, that rating is used to determine what problems are presented. Since I'm not paying attention to time, some of the problems seem to easy. Of course, these are the ones that, if timed, would require a solution within 30 seconds to avoid a rating penalty.
Tactics Trainer does a great job at determining how good you are at tactics, but the time pressure takes away from the "training" part, and if you ignore the time and/or ratings, the problems presented are not at the appropriate level of difficulty all the time.
Therefore, I recommend something similar to what was recommended in the OP. I would like to see all problems have a dual rating. One is the "timed" rating, similar to what there is now, and one is an untimed rating, in which the only thing that matters is whether you get the problem right or wrong. An alternative might be a "fixed time" rating, in which all problems are given three minutes to solve.
This would allow people who are using the tool for study purposes the chance to take all the time they wish, but to still get presented problems at an appropriate level of difficulty.
Perhaps have the tactics trainer rate you in several ways each of which you would need to click. For instance, you could have bullet classic etc... based on various game time constraints. Often people get them but it takes them time. True enough this should hurt the rating they get for the puzzle, but of what value is that rating then in 3 day per move correspondence chess? If instead you could toggle between viewing your rating in tactics trainer based on a variety of time based rules sets, you could get a clearer picture of where you sit within that rule set.