Option to use local Stockfish when requesting computer analysis

I'm really loving all the recent analysis features that have been coming out in the past months. Especially now with the ability to use a local Stockfish instance to analyze positions, we can look more deeply and accurately into variations than we could before with just the results of the server-side analysis. Because of this, it has become even easier to identify the occasional mistakes that the server-side analysis makes. I'm wondering then, how feasible would it be to have the whole-game analysis that is done when you "Request a computer analysis" be performed by local Stockfish, rather than the server-based Stockfish? It seems this would result in both a more accurate whole-game analysis, and a reduction of server load.

But why would you want to involve Lichess in this at all? If you want to do local analysis you can just paste the game in any GUI running Stockfish on your own machine, and start an analysis there. I also don't need Lichess to brew coffee for me...

In my experience, the server-side analysis is a lot more accurate (deeper) than the local analysis.

A problem with replacing the server analysis by a (deeper) local analysis of a user, is that you get untrusted input. Someone could send the server a lot of nonsensical evaluations so there would need to be some kind of server validation. And how would that happen? With an engine. This wouldn't reduce server load, but it would make things much more complicated.

@HGMuller It might be useful for the Chess Insights. (though I guess since lichess is open source one could just create a local Chess Insights program)
P.S. Are you the guy who wrote MicroMax?

@MoistvonLipwig - Indeed, micro-Max was my creation.

"Local" meaning local machine? How does Lichess know where to find your local Stockfish installation?