First try: precisely two answers entered, both 3 letters. Didn't feel very promising. Got back to it for a second try this afternoon and it all came in a rush. Saw what was going on with the problematic grid entries almost straight away, and by the second half that was a huge help in completion.
Took a short break and then went back to try and suss what the extra two possible members of the thematic group might be. Lovely PDM when I saw them.
I don't see anything particular to link to the wording of the Primes message, but that's OK - just pleased to have worked it all out, hope it isn't an actual race as I'll be trailing in after the medal ceremony is over and the press pack has gone home, and wholly agree with Enigmatist that the second of the two additional possibilities is well worthy of a place amongst all those in the grid.