My Saltarelle B/C box has what melodeon.net describes as the "modern" set-up, so I DO have a D on the push (along with the F on the pull.) and very useful it is! The "classic" set-up has another C on the push instead of that D, much less convenient. Hadn't heard of having 2 Fs together. The chord that I don't have and comes into a lot of tunes is Bm. Of course you can have any combination you like if you so choose to get your box modified.
I can now do quite a lot of chord playing for tunes in G or D, having both push and pull buttons for both: as predicted by my tutor, "that will come a lot later - maybe 5 years". And I do love using the basses alone as drones: will do that for some pipe tunes (if no piper about!)
Also if there a lot of melody players in a session, but little bass/rhythm stuff going on, it is possible to just play left hand only in tunes that work with the "three-chord trick", CFG, GCD, DGA, ADE, etc.
I also use left-hand block chords only to accompany myself singing.
And as a side issue and a promo for B/C boxes:
1. there are lots of R-hand chords you can construct, e.g Bb on the pull,
2. provided you know where all your buttons are and know your scales, you can play in any key. Eb anyone?
3. Lovely mellow low right hand notes, down to C below middle C, and the B below that ( "the foghorn"!)