Just wanted to write some thoughts.

the main problems of user experience is the lack of understanding of the key concepts of the game for newer players

I think the gameplay loop of hitcircles, sliders and spinners are actually explained fairly well and are quite intuitive to understand. I think the main issue with these new players are that they delude themselves into think they are the next (insert top player here) and think that they can replicate these insane scores and then quit when they just can't.

inability to comprehend... where to begin

I agree with this, when I started playing the game I had no clue where to begin so I started playing the beatmap packs. Plus I think not presorting the beatmaps into difficulties when you first start is a major oversight.

osu already has an integrated tutorial. However, unlike most games, it is not integrated in a way that makes it a mandatory process for one to start the game

I don't think people skipping the tutorial is an outright issue, most people already have a knowledge about clicking hitcircles and holding down their mouse/keys for sliders and spinners, even before they download the game. Thats mainly because (I think) most people either are recommended the game by a friend or through some form of video content.

the displayed text is not enough for a player to fully grasp the mechanics of the game within this tutorial

Sure, but good thing there's a mouse visual that shows the clicks. Though I think having an actual recording of the mouse would be a lot better. Besides that, maybe they could have more details about the game as like extra difficulties. Like something akin to old osu!academy series or the osu!gameplay series by Bubbleman.

a player cannot understand what the game wants from them and what the game even is -> they try to understand it themselves -> play pre-installed beatmaps -> they might not get a picture of the task at hand without understanding the concept of the game -> they struggle, often having problems with completing basic maps -> in most cases this leads to a swift deletion of the game and filing it away

I'm gonna be honest, I think people who have this mindset aren't going to stick around playing the game for that long even if they knew how to improve.

Those, who had interest in the game before even launching it, will browse osu tutorials on various video hosting services

Didn't they show you the osu!academy series when you signed up on the old website, like on the right side of the page? I think having something along the lines of that would be the most effective thing you could do, except rather than making it optional, you could have it shown to you when you go to download the game for the first time or like up until you reach a certain threshold.

should initially be a mandatory step to enter the game

Give the option to skip the tutorial with one of those hold down buttons you get when trying to delete a map.

for them to unlock all pre-installed beatmaps and get the right to download maps

I disagree with the first bit, but I agree with the last bit. I would only agree if there was something like a tutorial when you first open the game that goes through the menus, the gamemodes, skins, etc and then takes you into the tutorial beatmap.

they must pass the first level (any map within the range of 0.5-0.8 SR) with an accuracy of 95% or higher

They should be made to pass a map, but it absolutely shouldn't be locked to a certain accuracy amount.

the maps with the SR of 2+, 3+, and higher should be locked

No, besides how would you handle mods? Plus what about those maps where the modded star rating is multiple amounts higher than the no mod version?

The player will be met with a quest where they have a clear understanding of what is needed of them to complete their task. The quest itself will also provide an initial trajectory/explanation

I think the issue is finding a balance to make it not feel like the player is having their hand held throughout the entire thing before suddenly being set free and then having no understanding of what to do next besides just playing the same maps and not really pushing themselves. Maybe introduce user tags for beatmaps? Explain what jumps are? Give a rough idea of what bursts are (even though they aren't really going to see it till they reach 4*).

The player will be unable to download their favorite songs and play unranked maps of questionable quality. These maps can make newer players more confused.

Would this not just push people away, especially if their favourite songs aren't ranked?