Could just be coinkidink but the above solution worked for me as far as assigning duties goes, I also didnt pick up any of the Tholian drops either not sure that makes a diff but after 4 tries I was superstitious at that point.
I had finished the mission just fine on two of my characters, one from each faction. Upon the playthrough of my third character, I got the completion bug. I tried playing through the remainder again, but with the same result. I just tried to replay the mission on my first character, and despite doing everything as I did it the first time, it is bugging out for me now. I really don't think the assignment of NPCs has anything to do with it, as I am using the same layout as I used on my first successful run. I sincerely hope that if the problem is this widespread that the end date of the event will be pushed back at least a couple days. I certainly don't want to waste time playing it on the rest of my characters if they're just going to get stuck again.
Once you enter the base, before you open the door for Tasha and the others, change character, so you are in the character selection screen and reenter with your char, the misi?n will be ok.
Tried 5 times with all my toons, no probem so far..
I had the OP's stated problem on my Klingon Tactical toon. I tried 3 times to complete and it kept putting me either back in the tubes or the control room before you open the door. I did the above quote verbatim and selected the same toon at the character selection screen...played out the mission, and it completed. Thanks for the help and posted solution gaevsman, much appreciated.