TG is strict on this.
I have several times flown on a FRTW fare ( paid F class), but travelled in
C class on TG Asian routes where no F is available. I have requested
access to F lounge, and always been refused.

I was in the TG F lounge the first day the new airport opened.
On that day, at least, the C lounge was better equipped than the F
lounge. ( Hope its improved since then).

And I thought the TG lounge was the only Star lounge in the new airport -
are you sure UA has their own?