Check out this updated package:

https://nuget.optimizely.com/package/?id=EPiServer.Search.Cms

Great, thank you! Should I also install this (https://github.com/episerver/content-search-lucene)? When I try to run the Scheduled job "Optimizely Basic Search Content Indexing Job" I only get "Sequence contains no elements" so there's probably some configuration missing.

I think that is  optional, please refer to this link for documentation:
https://docs.developers.optimizely.com/content-management-system/v11.0.0-cms/docs/installing-and-deploying-search-service

I see that «11» is part of the URL, but it's linked from the CMS 12 documentation:
https://docs.developers.optimizely.com/content-management-system/docs/search

Hmm, in https://docs.developers.optimizely.com/content-management-system/docs/search they even mention that I should install Episerver.Search but that package has a dependency to Episerver.Framework >= 11 && <= 12 so that's a no go.

episerver/content-search-lucene (github.com) the code is now even open source so you can use just that.

the doc is out of date. I believe EPiserver.Search is used as a server while Episerver.search.cms can be used for the client. both can now be obtained from above link (.net 6/7 compatible) 

Stupid question; can I copy Episerver.Search.IndexingService and use it as a project in my solution or do I have to setup another site that's used as an indexing server?

I believe you can include indexingservice in your solution, and should probably do that as well. the reason is that if you set up another site for indexing server it could be complicated with licensing. disclaimer: I'm not an expert on indexingservice and it's been a long time since I last touched it.