It would be really could if the system analyzed every game while it is being played, and showed a quick summary of evaluation with helpful comments at the end. chess.com does it, it could be better for lichess. I wonder how much pressure it would cause to the server though.
I'd want to know how many players use the analyze feature for a majority of their games before considering the service to be automatic. If a large portion of games go un-analyzed now, automatic analysis would lead to waste.
I think the only reason chess.com can do this is because 1. they make lots of money and 2. The analysis that they do (for non-premium members that is) takes around 4 seconds or so of computing time, which is essentially useless because low-depth Stockfish is quite inaccurate. If you'll notice, it takes some time for the analysis to finish on lichess because it isn't doing the lowest possible depth.
It really isn't that much of a haste to click the analysis button yourself, and it definitely isn't worth the loss in efficiency.
I guess it would be heavy burden to servers and most of work would go to waste.
Players really wanting to improve should first analyse game by themselves, figuring out what was going on and after check with engine. And players playing for fun or fast "easy going" games don't need full analysis for all games. A simple (and inaccurate) brief summary like for example "White was losing but had a chance to draw and took it", but you need to analyse first to do so...
#3 I agree.
#4 It's an open question of whether there is some more cost-efficient way of producing an analysis of the same quality. To date nobody has found a technique which either makes multiple passes through the game and/or only deeply analyzes positions which require deep analysis to produce an accurate evaluation.
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