Does self-analysis of games even improve chess skills?

If you don't learn anything from your analysis, then no...it won't help. But looking at your games you should be able to see, particularly with the computer's help of what the better moves are... Where you did not make the best move possible.... But then it's up to you to understand why it wasn't the best move...

Unfortunately, to get really good at chess... Does require some work on your part.... Study..... Unless you're just an extremely gifted person. Studying your own games is a very beneficial part of that especially while they're still fresh in your own mind... You will see things that you didn't see during the game... Just like working puzzles if you see situations enough.... You'll remember them. For somebody like me that may take seeing them hundreds of times.... For a gifted person... A real savant at chess... They may only have to see them once.

Yes, for sure you should look through your games on your own, to see if you can spot the mistakes you made, and to try to come up with better moves that you could've played, instead.

Then, check again with an engine, to see what you missed.

You likely won't be able to remember everything or even to understand all of it. But if you even learn just one thing from each game - it's like a droplet of water into a bucket.

Eventually, you can fill a whole bucket with tiny drops.

You fundamentally misunderstood how the whole process works. The goal is not only to learn what to do in the exact same position. The goal is to spot patterns that will be useful in billions of different positions.

By your logic, practicing tennis is also completely useless. There is absolutely zero chance that any ball ever comes to exactly the same spot from exactly the same direction, speed and spin.

I find the game review feature to be of minimal value at my beginner level. Sure, the engine will point out a blunder, but I already knew that I had blundered. When the engine, which is around 3500, suggests a best move, it isn't usually helpful, because as a 1000 player, I am never going to find that solution. Example: in a recent game, I blundered, leaving a hanging knight. The engine, however, said the move was "brilliant", treating it as a sacrifice, not a blunder, that eventually - eventually! - would result in a winning position. In the game, my opponent, also a beginner level, jumped on the hanging knight and proceeded to win the game.

Yes, self-analysis of your games is one of the most useful ways to improve.

I agree that the computer Game Review is not very helpful. It can tell you if you made an opening mistake or a mistake in a tactical sequence.

However, it does not EXPLAIN why one plan is superior to another. It does not identify the aspects of the position that make one plan better. This is the kind of understanding that distinguishes an advanced player. And this is what you should try to do when you analyze a game

Also, learning is most effective—faster, deeper, more lasting—when the learning process is active and meaningful. A player who spends 30 to 60 minutes analyzing one of their tournament games will learn more than another player who looks at the Game Review without trying to understand the reasons for the suggested moves, or another player who spends the same amount of time playing speed chess or trying to memorize an opening sequence.

Top-level players routinely review their games to understand both tactical and strategic errors. It’s a proven method to identify weaknesses and refine strengths.

None of the famous players, not one, got to the place they ended up in without severe studying.

Not Bobby Fischer, not Garry Kasparov, not Magnus Carlsen, not Paul Morphy. Nobody.

No "natural talent" can give you a shortcut to studying. To be good at chess, you need to know a lot of chess patterns. Scientists estimate that top chessplayers spent about 50,000 hours on learning chess and learn about 50,000 chess patterns in this time.