Rex had to be very careful to avoid being killed.
Without going into full details, in Clone Wars S06E02 , clone trooper Fives discovers inhibitor chips, in S06E03 he manages to remove it from his head with quite complicated surgery. In S06E04 Fives is directly told by Palpatine some parts of anti-Jedi conspiracy. We never know how much, because this is not shown on screen. Fives manages to tell Rex about chips, that conspiracy against Jedi exists, and that Chancellor is involved. After that, Fives is killed, ironically by other clones.
Considering what he saw and heard, Rex had to move very carefully. He first had to check if he really has inhibitor chip, then find medical droid willing to extract it discretely. Rex had to be cautious to avoid ending up like Fives. Therefore, he only told about chips to trusted friends, because other clones, obedient as they were, would probably report him in and he would end up killed. Note that Kamino cloners were forced to admit existence of inhibitor chips to Jedi after discovery made by Fives. But they invented cover story that inhibitor chips serve to control overly aggressive behavior of clones, and that removing them would make clones go mentally unstable (like Fives supposedly did) . So Rex removing his chip publicly would probably got him in position of being declared insane, and therefore target for elimination.