Though they were both played by the same actor (James Lashly), Primmin was not on TNG. His TNG character was named Kopf (end credits, I think). Plus, Primmin told Odo he was in security for 6 years in "MOVE ALONG HOME", which means he would have been a security officer since season 1 of TNG. His other character in "Brothers" was early season 4.Ben the Waiter or Mott the Barber... Mott would have largely overlapped merchants like Quark and Garak as a gossip agent but at least we would've provided a much needed splash of color.
We did get another TNG character that didn't work out, Lt. George Primmin. He was only in two episodes in S1 but was an Ensign in TNG.