Unfortunately, that would mean that W only gets to play on 4 teams while everybody else gets to play on more and six players (A B C D F O) get to play on 7 teams. I doubt your players will consider that fair.

Update: You can get pretty fair with this subset of teams:

A, E, F, G
A, H, I, J
A, N, O, P
A, Q, R, S
A, T, U, V
B, E, H, K
B, F, I, L
B, N, Q, T
B, O, R, U
B, P, S, V
C, E, I, M
C, G, K, O
C, J, L, P
C, Q, U, W
C, R, T, X
D, E, J, N
D, G, H, L
D, M, R, V
D, S, T, W
E, P, Q, X
F, M, S, U
F, V, W, X
G, I, N, R
H, M, O, W
I, K, S, X
J, K, Q, V
L, N, U, X
[download]

Which has half of the players on 5 teams (A B C E I N Q R S U V X) and half of them on 4 teams (D F G H J K L M O P T W).

Update: And if you add the following teams:

T, K, G, M
D, F, H, P
W, J, L, O
[download]

Then every player gets to play on 5 teams and there are only 4 pairings that ever get repeated (DH KG WO JL).

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