In Wimbledon Why Don't They Let You Keep A Tennis Ball If You Catch It?

They do not get to keep the ball because it is against the history of  England.

Unlike the 2009 U.S. Open, in which spectators were permitted, for the first time,
to keep balls hit into the stands,
most tennis events follow the International Tennis Federation (ITF)
and/or United States Tennis Association (USTA) or similar rules
concerning the number of balls allowed to be in "play" at any given time (usually six).
If one of those balls goes into the stands,
it is expected that the spectator who retrieves it throws it back,
assuming the players are not preparing to serve
or aren't already playing the next point,
because, if something goes wrong with the other balls (e.g., decompression),
then they have a "back-up" ready.
Since it is rare for balls to be "lost" or otherwise be taken out-of-play,
the U.S. Open believes that they have enough of the current six
to last until the subsequent ball change or the match ends,
so they allow spectators to keep the ball.
IMHO, that's a risky "bet", because if all six balls go into the stands,
then they have to get all new balls, which changes the tenor of the game
(i.e., other players in other matches would want new balls,
so they might purposely hit all the current set of balls into the stands!
Of course, this might be considered a code violation for ball abuse,
so the player would have to hit them into the stands inconspicuously!).