A privacy focused client that offers personalized recommendations is a bit of a juxtaposition IMO, and is definitely impossible now that YouTube removed homepage personalization for logged out accounts.

If you are fine with YouTube without ads, then you could log in with the likes of Revanced or Grayjay, and get a feed with personalized recommendations. This isn't private though. Grayjay does however have a feature to anonymize your watch activity, and to perform certain actions in an anonymous YouTube session.

I'd even maybe be fine with something that takes all the watch next recommendations of the videos in the subscriptions page and jumbles them together as "recommendations"

NewPipe (and Freetube the last time I checked) already does this, but they use a built-in subscription system that does not require any kind of account

Piped (and LibreTube) uses its own account system which allows you to sync its built-in subscriptions & playlists between mobile and desktop, but I have not looked at whether they offer a feed with the synced subscriptions.

Personally if I'm specifically after watch recommendations, I just look at the related video suggestions on something that I like