Going back to someone's earlier post on Nethack and other roguelikes with a hunger mechanic - are those games suddenly terrible?
If there is a strict limit on food availability? Yes. I don't like most rogue-likes for a variety of reasons, but tack on some kind of time pressure and suddenly they are 10x worse.
What about racing games, where players must go faster (and thus achieve a better time) than other racers?
Racing games don't really fit here. Trying to go faster != being forced to play under a time limit.
Is the Super Mario Bros series awful because of the timers? I certainly don't recall the timer ever giving me too much trouble.
If the timer never gave you any trouble, why have it? Seriously, why have a time limit if it is generous enough to not be an issue?
Keeping track of time for score reasons may be fitting for some games, but there is no need for a
limit.
What would be a good alternative to a timer?
Consider Warzone 2100. It has a persistent campaign, and all the levels have a time limit. Generally these limits are pretty generous, and they really exist to forcibly keep you moving. This is a crude and obnoxious way to keep the player from "breaking the game" by farming an easy map for unlimited resources and building a gigantic supply of power and units.
In this case a simple power and unit transfer cap would be all that is needed. Fluff it with something about the transport only being able to hold so much or something similar and it wouldn't even seem weird.
Look at STALKER. In the first game some of the missions had a time limit. I always found this hugely annoying, I could never just play, I always had a little worry that I wouldn't make it in time. There is a reason the third game is my favorite by far.
Look at any Bethesda RPG ever. No time limits, but dialog certainly encourages you to hurry anyway. If you have a reason
not to hurry then you don't have to. You never have to worry that if you don't travel quickly enough something bad will happen. If you want to check out something on a hill nearby just go do it, no worries. Sure, a generous timer would allow the same thing, but then you have this vague "time anxiety" leaning over everything you do. Yuck.