Yeah, when I think time limits, I think more Chip's Challenge. I miss "hardcore" puzzle games.

From what I can tell, that seems to be similar to how the Touhou games handle it. It's also much more fair to the player than taking away all levels on death.

In a way, the mechanics serve a similar function in these sort of games, as they prevent the player from being able to play forever without making progress (except maybe bosses in autoscrollers, though some of them (like the Touhou series and I believe Crimzon Clover) have a mechanic where, if you take too long to clear a boss phase, the game skips it and goes to the next phase (which makes a pacifist run possible in some games).

As for why you might not want the player to play forever on one life:
* For games in arcades, the money earned by the arcade operator is based off the number of games played. If a single player can play forever on one credit, the operator doesn't get any more money. There's also the fact that nobody else gets to play on that arcade cabinet during that time (assuming single-player).
* This sort of mechanic also keeps the game's scoring system meaningful, preventing players from accumulating unlimited amounts of points in one life. For the hypothetical game design being discussed in this topic, it also prevents the player from being able to just stay in one spot and farm XP until they've broken the game. (Allowing the player to do so is reasonable in a conventional RPG, but this game is not a conventional RPG due to its structure.)

All fine except the autoscroller, if it's only a section it's ok though. Autoscrollers are dumb and belong to Android crappy games :) As long as the enemy numbers are not limited and/or it's possible to speedrun without grinding. Like my previous response, if there's no artificial limits imposed and dying only means loosing time/ increased dificulty otherwise it adds nothing. The implementation is far more important than the ideas, unless it's a very creative game.

Sounds a lot like Smash TV.

I wouldn't know... nowadays Turrican would probably be considered a Metroidvania. It came to my mind because of the time limit.
I knew this games by heart at some point - I only ever died of time limit and had ~40 lives left in the end. I also caused the score to overflow almost every time (max for hi-score was iirc 999,980, at 1M it went back to zero).
Losing a weapon level wasn't a big deal either, since the levels were pretty well stocked with upgrades, and only the basic was pretty weak.
PS, regarding time limits: Have you ever played The Killing Game Show?

This thread made me remember Cadash, and that is good.

No, haven't heard of it.

There were many games following a combination of these or similar limits back in the day.

For example:

*) Stage-based? Check.
*) Stage time limit? Not directly per stage, but there is a time limit of 1 hour to finish the entire game.
*) Experience and levels? Nope.
*) 3 lives? No, but you had 3 hit points so you could be hit twice before dying. You have infinite lives but every time you die (which can happen very quickly since traps and big falls instakill you) you get sent back to the beginning of the stage. So you need to git gud or you won't beat the time limit.
*) Short? Check. You have to beat the game within 1 hour.

Can anyone guess the game?

I actually did think of the game sometime after making the topic, but there are a couple differences:
* The time limit is global, rather than per stage. (You do get extra time at certain points, and there's a spell that can get you more time.)
* The game doesn't give you multiple lives unless you insert more coins, and it appears that you don't lose your experience or items if you do continue.

(Note that I have not played Cadash, but I've watched speedruns on YouTube.)

One of the classic (2D) Prince of Persia games.

Bingo. :-)

I knew this wouldn't take long since people here know their classics but that was guessed correctly faster than I thought.

I distinctly remember reading about Prince of Persia in Nintendo Power, and I distinctly remember the mention of the time limit, which made me not interested in the game.

(I also remember seeing mention of some way to skip most of the game.)