Takei actually was a regular, as was Whitney. Doohan wasn't -- until season two. These things were contractual.
Ah ok that's interesting. I'd assumed that the position in the cast list had some bearing on contracts. Although either way, only Shatner and Nimoy were guaranteed appearances.
Of course Chekov is missing from this episode so I suppose the only appearances we have of all the 'gang' appearing in the same story are TMP and STIV, although there are probably a few episodes in seasons 2 and 3 featuring everyone but Rand.
Plus there are the recurring characters who didn't make it to the movies like Kyle (apart from ST2) and Leslie.