It's impossible to prove every adverse omission is racism-based, though there have been subtle and not-so-subtle TOS incidents which suggest the possibility of it.
First, in the not-so-subtle, there's the sudden promotion of DeSalle to fifth-in-command in CATSPAW the very moment Uhura would be the natural fifth. DeSalle is the only non-regular Enterprise crewperson ever to sit in the center seat in TOS. It's only his second, and final, appearance. And suddenly his shirt's promoted from yellow to red. Quite convenient. I sense major placating of the South here.
It was DeSalle's third appearance. He was in season one's
"The Squire of Gothos" and
"This Side of Paradise," making
"Catspaw" the third (and final) appearance.
The fact DeSalle made it to season two seems to suggest he was one of the 1st season guests thought to have some sort of potential, so he was worked into the "
Catspaw" script. That the character assumed command over Sulu in this episode suggests no bias or skipping over him for the position, since Sulu held temporary command a season earlier in
"Errand of Mercy" (and during a tense situation normally reserved for leading characters).
The next two points are subtler. Takei and Nichelle Nichols were the only two of the key seven in THE WRATH OF KHAN to lack significant scenes or moments. And, if were George Takei, I might even wonder why of all the six major male players, I/he alone was never temporarily killed in TOS.
Let's not forget Takei was supposed to have his promotion to captain scene in TWOK, but the scene was cut. That was major for one of the "B" team, and certainly proof of how his character was viewed behind the scenes. Moving to TUC, Sulu has his own ship. How often do "B" level characters ever receive that kind of promotion? It is rare.
Moreover, on TOS, Sulu was in the captain's char on several occasions, which would have never found its way in front of cameras if the unsubstantiated racism had any merit.
Regarding the topic, as noted time and again, one cannot charge racism as a cause about a specific situation, then fail to offer equally specific evidence about the event & players involved. No conclusion can be reached from false premises dipped in a pool of inflammatory language.
Further, to ignore relevant examples (the scene in
"The Way to Eden"), which for 1960s TV counters any notion of Takei facing true racial discrimination is dishonest. Particularly when considering the woman who shared his scene was white. Again, this was 1960s TV.
One cannot allow an agenda-gripped mindset leading to a search for evil that did not exist in the production offices of TOS..
...otherwise, someone would have found it easy to produces something--
anything of substance which supports the charge.