Nothing so nice will be possible if there is not a multiple of 4 players. The schedule you give for 24 has the property that no two players are on court together more than once, and some players do not meet. If that's what you want for 22 players, then the best that I can do is 6 rounds as follows:
( 1 13 11 16) (12 19 6 10) ( 5 2 15 20) (18 14 8 9) ( 4 17 3 7)
(18 16 17 5) ( 8 2 13 7) (22 12 14 3) (15 1 4 10) ( 9 21 11 6)
( 8 11 5 10) ( 7 1 14 19) (13 20 3 6) (15 16 21 22) ( 9 4 12 2)
( 9 13 19 17) ( 3 18 10 2) ( 4 20 11 14) ( 6 5 22 7) (21 8 1 12)
(15 12 18 7) (16 8 4 6) ( 5 9 3 1) (19 22 2 11) (17 10 20 21)
( 2 6 14 17) ( 3 19 8 15) (21 4 13 5) (18 1 20 22) ( 7 10 9 16)
Players 11 to 22 have a bye and so have one less game than players 1 to 10. It might be more sensible to look for an 11 round schedule where everyone has exactly one bye, and some players play together twice, but I don't think you want that?
