One problem I see with just analyzing your games with an engine is that the engine gives (what it thinks to be) the best move, but doesn't explain why it's the best move and, more importantly, how should I have spotted it.
Sometimes it's very obvious why the move (or moves, if you turn on multi-pv) it suggests is the best, but it's still not always obvious how I should have seen or noticed it. Sometimes the strength of the move only becomes apparent after the game progresses a few moves ahead, so it's not necessarily so obvious to see at that point why it's the best move.
Othertimes it's not clear at all why a move is the "best" move, especially if the evaluation barely changes. In these cases engines just don't explain why. Other than, perhaps, with an implicit "all the other possible moves are worse".
Conversely, an engine may show a move of mine to have been a blunder (if the evaluation drops significantly) but, once again, it doesn't necessarily explain why it is, and it might not be immediately apparent why. It might not even become apparent even after a half dozen moves. The engine just calculated with its sheer brute-force power that 20 moves ahead it leads to an inferior position, but that's hardly useful for me.
More useful would be to know general principles and tactics that improve the position, ie. a higher level explanation of what should be done in a particular position and why (ie. not a "20 moves from now your position becomes better by a complicated evaluation function", but a "it strengthens your position by defending these pieces" or "it weakens your opponent's positions by putting pressure on this piece" or similar.)
If I'm reading my chess.com statistics correctly, during the last 3 or so years I have played 3970 games. Pretty much during all this time have been stuck at about 1500 rapid rating and 1300 blitz rating, with no discernible improvement. (I haven't been playing a whole lot of rated games, but I don't see myself winning disproportionately many games against people at my level in unranked games either.)
Is this normal? Does there become a point where merely playing games will not help you improve at all?
I've been on this site for 7 years and my blitz rating and rapid has not improved. I did improve in bullet and puzzles though