No, "Gamesters" was a Desilu episode. It says Desilu in the credits. I repeat: "The Immunity Syndrome" was the first episode in production order (with the exception of "Bread and Circuses") to say "A Paramount Production", while the first such episode in broadcast order was "A Piece of the Action". The last Desilu episode to air was "A Private Little War".
I said "with the exception of 'Bread and Circuses'" because that episode was 14th in the season 2 production order, but for whatever reason, it was the next-to-last episode to be broadcast that season. Since it took so long to make it to air, there was enough time to revise the end credits to say Paramount instead of Desilu.
As for that "tall peak" Paramount logo: As I said before, I'm pretty sure that TOS reruns in the 80s kept the original logos on season 3, because the copyright was on the logo. For the most part, season 3 logos weren't altered until TOS-R, when CBS replaced Paramount's logo with their own, and transplanted the copyright to their logo (and, inexplicably, stuck the DESILU logo on at least one S3 episode, but that's another story).
And finally: To my knowledge, the 1969 logo, the one you'd have seen on The Brady Bunch, never EVER showed up on TOS. A friend of mine *remembered* that logo showing up on local reruns, but since he had other false/distorted memories, I'm writing this off as one of them. (It also wouldn't have been attached to the end of Desilu episodes - I've seen a recording from WPIX in NYC when they were still running 16mm prints, and it's just Desilu, with no added copyright.)
Anybody still got recordings of TOS from the early 80s, when they were still being run from 16mm?