Part of the reason for having a set time limit, i guess, is because of the way the problem rating works. If you solve a problem then your rating goes up, but the problem rating goes down and vice versa. If you removed the timer it would be unfair on those who still had the timer because the problems with the same rating would be harder for them than those people without a timer. As you say it is fun to quantify achievement relative to others, but having an option on time divides the userbase in two.
That said it wouldn't be that hard to implement a two rating system, one for timed and the other not.
I skipped through some search results and obviously a lot of people, like me, are unhappy with the tactics trainer point system. Time usually is too short to really calculate your moves if you'd like to improve your rating at the same time.
I have failed to find a satisfactory answer as to why the system cannot be changed. The only regular answer that makes sense seems to be: "Ignore/turn off the timer, ignore your rating, and eventually you will make progress" (because while you will still not be able to calculate the moves beforehand, you will recognize the patterns and most time guess correctly).
While I can respect that position, it seems pretty crazy, too. Why should you have to ignore a part of the tactics trainer that is meant to encourage and motivate you? Almost any computer game does include a rating of some sort, stats you can improve, levelling up, and so on. Why? Because it's a lot of fun to reach new limits, being able to quantify your achievement in some sort, of course! And as to patterns being the only thing that matters, I beg to differ. Any game will include positions that are new to you and will require you to calculate.
So if anyone could enlighten me as to why the point system cannot or should not be changed, for example in the way I have sketched out in this blog post, please do.