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Should people give up early?

Ive recently joined Lichess.org, cause I saw it on yt saying it's all open to players who want to learn. After playing rapid mode too much I realised that mostly players were making wrong moves like hanging there queen, etc. and they'd not even try to request for a takeback. So they do this cause they are ashamed or afraidnto ask for ?
Well, takebacks are usually only used for obvious mouse slips (if ever). It is not the purpose to create a game free of mistakes.

If you hang a piece, just live with the consequences, and try to be more careful in the future.

At your current rating, hanging pieces left and right is to be expected, so enjoy it while it lasts.
Should you give up early?
Or
(Why) Should you not give up early?
1) If you are someone who has taken up like 1 hour of online chess in ur daily routine, or just play for a nice amount of time, you should NEVER give up early. Lost a rook? No problem, if god is on your side. you'll get the rook and the queen, and if he's really grateful, he will give you the game itself. Are you down some pawns? Again, not to make a huge fuss about it, your oppn. can just commit a mistake in the endgame. Do remember, these things won't happen always, but they sort of have 50/50 chance of happening, so be optimistic and not a pessimist.
2) Are you a person who is working? Have a job and like to play chess in your tea break? Then feel free to do so, and even feel free to just resign a game as soon as you are down material. Generally most of the people who have such jobs and have to work 10 hours a day don't really worry about their rating, and since they can't really pursue chess as a career anymore, they just take it as their passion/hobby. So you can resign the game whenever you feel like it or if your tea break/lunch time is over.
Sometimes I give up after losing a pawn. I know I shouldn't but I don't wanna play a game where I have 90% chance of losing it would just frustrate me and the mistakes already happened so there is something to be learned. I wish my opponents did the same but they play till the bitter end.
Also, one more thing, before writing all of this up, I only saw the title and not your post. Obviously, one seeing the title would firstly think about resigning the game too early, and not about giving up early regarding takebacks. If this helps, then take whatever you can from it.
Sorry :C
A lot of people have takebacks disabled anyway, so everyone gets used to not having it.

I had the opposite experience. People would hang a piece under a lot of pressure and then beg for a takeback. (In a rated game? Online players are such Philistines sometimes...) When people realize they can get takebacks they abuse the snot out of it. A lot of Lichess players even want the feature removed entirely, but I think the option of disabling takebacks precludes this change.
I play bullet a lot, and in many rare cases, my opponent offer me a takeback. Me? I get a bit too sly. I just wait there till his time runs out; cuz he's sitting there thinking that I will accept the takeback when in reality he's just losing time.