There's a whole spectrum from instant re-spawn to permadeath.
Side-note: in COD it depends on the game mode, and instant-respawn modes stem from the early FPS games where it was standard.
In my recollection, Counter-Strike popularized the 1-life-per-round team vs team type system within FPS games. It existed in others before that, but CS very suddenly became the most popular online FPS, making it a common standard.
In the CS system, dying can result in up to a 5 minute wait, while you're left to spectate the rest of the battle. This puts a much larger emphasis on playing carefully rather than running around like a headless chicken.
CS (or some COD modes) is also focussed on a match of skill between two teams trying to complete/prevent an objective. It's supposed to be (somewhat) tactical game with tension building as the player count on each team dwindles. If a team that's down to 2 players comes back to win against a team that still has 5 players, it's exciting. The limited lives per round is a source of drama, as well as a behavioural modifier.
If it had instant re-spawn, it would be a completely different kind of game, so this is like asking why every game doesn't have vehicles and rocket launchers and jetpacks!
Being frustrated by dying in that kind of game is by design. You're supposed to regret and reflect upon the choices that led to your death, and you're given the opportunity to spectate the fate of your team-mates. This in turn affects the behaviour of the players, so a COD player in a deathmatch mode will behave differently to a COD player in a one-life team mode.
If you can't deal with idle reflection for 10 seconds to a few minutes, then don't play that kind of game.
Seriously, I do not understand at all when I play COD and see people raging about a 10s re-spawn timer. I just assume these people must have ADHD
In a quake/deathmatch game on the other hand, the idea is to constantly be as active as possible, being on the move and always attacking, in order to get your kills count up as high as possible as quickly as possible. Time you spent dead is time that other players are increasing their score, while you're not, so you want to respawn as fast as possible and get back to it.
Increasing the respawn timer here acts as a penalty for losing a battle, which leads to a score feedback loop -- winners don't die, so have more game time per match, which gives them more opportunity to gain points / losers are penalized with less time per match, which gives them less ability to gain points. These kinds of feedback loops create more contrast between different skill levels on the scoreboard.
At the other extreme, in a perma-death game, you might play a character for hours (or days, weeks, months) and when that character dies, they don't come back. This is a completely different kind of fun, and results in completely different kinds of behaviours. Players in this game are not going to act like Quake/COD players -- they're not going to run around in the open, guns blazing, hoping they'll head-shot, their enemies before they get shot themselves. They'll slowly crawl from cover to cover, using binoculars to see if anyone is hiding in distant windows, spend minutes scouting areas to ensure it's safe before revealing themselves, etc... and when combat occurs, they may even feel real fear due to the real possibility of losing something. They may even empathise (that's not a word you associate with COD/CS/Quake) when they kill someone, knowing they've just erased days of work! That ability to evoke real empathy and emotion is not something that a meaningless deathmatch game is capable of (in my experience), so they're each a different kind of art, made for different purposes, aimed at different demographics.