The Barre F is WAY overrated.

I know classical style players have to use it; I certainly did when I was a classical player. But in every other music, you need to find a chord shape that lets you move quickly and smoothly to and from F.
What's that? Mostly, it's the 4 string version: xx3211 (low E string to High E). And occasionally put down the 5th string at the 3rd fret to give you the C note for a fuller sounding version.

And someone suggested using 4 string jazz chords, the kind where you mute some in-between strings.

Wonderful idea; use everything you can learn from all types of music.