That seems unusual at this point. I’d have figured that there would have been some movement to have a crossover with these properties. I suppose you’d need some expendable red-shirts (for TOS), as well as non-essential personnel for the other Trek era teams. An additional villainous group would also help the drama (Klingons during TOS, also Romulans, Cardassians, others).
Also, to make the threat more palpable, it would probably take disabling or removing entirely the resources of a starship (e.g., beaming people away or the monsters).
How would the leadership differ between Kirk, Picard, Sisko and the others?