"I have no idea why this site thinks that you learn better this way, when mine and most others' ratings have been stagnant over the years."
Where did you get this from?
The idea of this tactics trainer is to quickly recognise and build up lots of tactical patterns. If the average time is 15 seconds it's either an easy hanging piece or an obvious pattern that you haven't learnt well enough. Being able to spot them quickly isn't just good for blitz but for recognising possible opponent attacks easy, ruling out candidate moves, knowing where to put your pieces. Knowing basic patterns well allows you in a game to build towards more complicated lines, and meaning you don't have to calculate as far.
If you want untimed try chesstempo or a chess puzzle book. This isn't designed for that. Or just take your time and you will progress as you start to recognise positions more quickly.
Plenty of patterns I see as soon as the board loads and it's a case of just checking for defensive resources.
If you're complaining that there's always a clearly winning move, that's the idea of tactics trainer. There are enough find the best move programs if you don't want it to just be tactics. (Chess Hero, Find the Move, Chessgames Guess the Move)
On a recent puzzle the average time was 15 seconds. That's absurd. What can you learn from a game you're not really playing and only have that much time to make a move? If I was in on this game I would know how many pieces I was up or down and many other important things that take up time. Like which side I'm even playing!
If I take any time to really examine the board that was just thrown in front of me, I'm going to lose points to those that make quick moves of reflex.
Tactic trainer doesn't really teach you to think. Where am I going to find myself in a situation where I run across a random game and have to quickly make the best move possible?
Plus, people may have distractions at home. Doorbell/phone rings, etc. The timer should be taken out completely (for scoring purposes) since solving it correctly matters the most. Being rushed to solve it, leads to you making thoughtless moves, and what do you learn from that anyway?
Everyone ability to make the best move isn't going to skyrocket if everyone can be allowed to think at their own speed. Their ability may increase because they are allowed to think about the problem. The ratings for the problems would remain relevant.
I have no idea why this site thinks that you learn better this way, when mine and most others' ratings have been stagnant over the years.
So no, I don't find myself jumping into random games I've never seen before, and need to find the best possible move in a matter of seconds. The tactics Trainer should be helping you think tactically. In a learning environment, you shouldn't be getting rushed. That's for actual game play if the timer does become an issue. (I mean seriously, would you expect a video lesson to have someone talking as fast as possible with no replay button, and if you didn't catch it, then too bad?)
I can spend several seconds on: figuring out what color I'm playing (something that has rarely hurt my actually games), examining the board, and only then begin to think about a tactical move. With 15 seconds average time on a clock, I might get it in time to score a positive rating, but only because the problem was so easy that I learned nothing from it.
If anyone want to argue that it's a reflection of how other players did, no, it's not. A higher ranked player would knock the time down further, but then they aren't given these problems. These are given only to a subset of players at that particular artificial TT rating.
Knocking the timer out of the loop can only help people think more, allowing them to grow mentally with their skills in solving these problems.