Competition is all about adrenaline.
You've probably experienced a competitor's high. It's that feeling you get when the basketball team your cheering for hits a last second shot to win the game, when you run faster than you ever have before, when you beat your previous best score on a video game, or even when you complete a difficult puzzle. For competitive people, this happens even in weak escape rooms, but a good escape room can produce that feeling in even uncompetitive people. They can do this by creating an immersive environment with a thrilling theme and an intense soundtrack. They can do this with many simple puzzles that create a feeling of momentum or with a few puzzles that are very challenging. They can do this by giving the team a goal to shoot for such as a number of hints used or time passed. I played one room that you could actually escape without completing all the puzzles. If you were really good, you could open the final lock with the solutions to three of the final puzzles, or if you needed more help on the final lock, you could easily open it by completing all five of the final puzzles.
A few times I've seen game designers attempt to create competitiveness by limiting the roles of individuals. It usually happens when only one person on the team is allowed to enter an area, hold the flashlight, write on the notepad, enter a code, or some other action, but I've never liked that kind of limitation. For me that has actually removed the competitor's high.
I feel the same way when designers make a room so dark that you can't see the puzzle you're trying to solve. I understand that darkness adds to immersion in some scenarios, but making a puzzle harder by limiting a person's ability to see the puzzle will do nothing but frustrate players unless turning on the lights is actually a puzzle that needs to be solved. I've often seen designers give a team one light source in order to make the room more linear and make people work together. That is not a terrible mechanic unless it goes on for too long. I only like this when the team is challenged to use that single to turn on or discover other lights. More often than not, I think low lighting is a way to hide a poor room design.
For competitive people like me, there's nothing worse than a poorly designed puzzle that creates unnecessary limitations, omits necessary information, penalizes players in a way that does not fit with the theme, or is too difficult to complete without excessive trial and error. I'll also add here that it drives me crazy when a game designer gives no clues about what lock the solution to a puzzle is meant to unlock.
This point is not meant to say that the goal of a room should be competition, but the adrenaline that competition adds makes a room more fun. In fact, in a good room the sole purpose of competition is to add to the entertainment. There are some legalistic game designers and game masters that are all about the rules. The competition is so important to them that they forget their primary objective is to entertain.