(Standard boilerplate, not specific to this extension)

The first step of https://www.mediawiki.org/wiki/Review_queue is

WMF stewardship: Find at least one WMF team (or staff member on behalf of their team) to agree to offer basic support for the extension for when it's deployed to Wikimedia Production.[1] Do this step before any other steps. Due to technical debt, WMF capacity to steward extensions is limited.

Moving on to specific extensions:

I have no idea who want to steward this.

I also doubt this would solve the intended use case - Google Translate (since I don't speak French) of the linked discussion says "Anonymous users always see the same books on the home page (for several months), while the books change every day for logged-in users." This sounds like the problem is the https://wikitech.wikimedia.org/wiki/Varnish caches, which are upstream of MediaWiki and MagicNoCache (even in the unlikely event that it gets installed) probably won't solve.

Probably the best approach for the actual problem is to run a bot to purge the main page every day - enwiki is doing that every 15 minutes IIRC.

Please install the MagicNoCache MediaWiki extension on French Wikibooks site.

Which specific (!) problem would you like to see solved? T351573: Main Page offers yesterday's content for logged out users?
This ticket describes only one possible solution but no problem. There might be better solutions but that first requires knowing the problem.

The problem is main page for non logged-in users is stalled for multiple weeks, whereas content change every day (at 00:00 local time).
The best solution would be to reduce cache duration to 12h or less.

That's tracked in T351573: Main Page offers yesterday's content for logged out users and potential solutions should be discussed in T351573 to avoid fragmentation of discussions in several places as that makes finding the best solution harder.

Installing MagicNoCache should be done to try this solution.

Task should not be declined on a simple "doubt".

Wrong place to discuss the best solution for the problem duplicated into this ticket. See my previous comment.