It would have been just as easy to have cast the same people but as totally new characters with a new ship, etc.
No it wouldn't. Why do people keep saying this?
"Hey, let's do The X-Files,
but without Scully and Mulder! That's sure to work!"
Granted, it worked for TNG, but it was definitely a gamble. And we'd been waiting nearly two decades for a new TREK tv series . . . .
(Not counting the cartoon series.)