Yes, that was a case where the fact that both characters were recast actually disguised a major plot hole in the film. If Connery had stayed in the role of Bond, I'm sure they would've had to address that in some other way.
The makers of OHMSS actually considered saying that Bond had had plastic surgery to change his appearance at one point, but they decided against it. It was probably a good decision not to call any more attention to the casting change than necessary.

...But the movie explicitly shows us that Spock physically recognizes them both. He identifies both Kirk and Scotty by name a second or two after seeing them. Why would you discount a simple, clean explanation like that in favor of something more convoluted like "Spock saw something that reminded him of the TOS Kirk and Scotty, but it wasn't their physical appearance"? I get that you don't like the Kelvin movies, but it's strange to outright reject what the movie plainly shows us. I just don't see any advantage to that.