I think the various Star Treks do try to keep some sense of recurring extras. Voyager did this as much as they could due to the premise of the show. TNG tended to stick to a handful of day players for the helm position and various other background crew. DS9 had Morn. Discovery's bridge crew has more or less stayed the same since the beginning, existing in a weird state somewhere between day player and actual cast.

It's also not a fixed crew, other than Voyager and Discovery because of their unique circumstances. Realistically, most of the crew would only serve a few years at most at a single post before being transferred and/or promoted somewhere else.