Primeape is a lot better now that Sawk's been banned, since it lost its biggest rival.
Comments like this really grind my gears. Primeape didn't magically get better because a superior mon that you state fulfills the same role (which is doesn't) left the tier. It just means that Primeape is one of the only fighting types left in the tier, so if you want one, it's one of the few options. However, at this point, Hitmonchan is probably the best "offensive" fighting type left considering it has access to priority, hazard control, and isn't piss weak without choice band. In addition, Combusken, Poliwrath, Hariyama, and even Throh can do so much more to support a team that I find it challenging to justify using Primeape ever.
Regardless of the concept of "who is the best fighting type", the meta was so focused around the need to have a fighting type before because the ones we had were really fucking good (Virizion for a while, Sawk, Gurdurr). Primeape is still pretty mediocre because it's really weak without band and it's real niche is in the encore set, which honestly isn't too useful since you don't have the opportunity to encore much in this meta and it's speed tier is mediocre for an offensive poke with such little power.
Primeape does NOT support a team at all. It's defensive typing is really mediocre and it's not a good switch in to knock off since it needs a boosting item to punch holes, unlike Sawk which could hit hard regardless. U-turn is meh in a tier full of Rocky Helmet Garbodor and hazards in general. Defiant is way too situational as most teams are utilizing a spinner like Hitmonchan or Claydol in conjuction with Xatu, forgoing defog at all. And the common defoggers (Skuntank, Peli, Shiftry) are not mons you really want to switch in on anyway because you're so frail and probably don't even live single attacks from and in the case of Pelipper can't hit back hard anyway without stone edge, which is so weak otherwise.
It's just a fallacy that one Pokemon of a certain type leaving makes other pokemon of the same type better. Primeape is still just as underwhelming as it was when Sawk was in the tier, it's just one of the few offensive fighting types left (Although it's still significantly worse than the likes of Combusken, Hitmonchan, Poliwrath, and Hariyama). The meta has shifted from the need to have a fighting type on every team and this is a healthy shake up.
The logic you used in your post is flawed and should not be used to give buffs to Pokemon's ranks in this thread. Here's an example that may make this more clear to you:
Suppose Xatu got banned, Mesprit, Musharna, Claydol, rose via usage. You wouldn't then say, "Guys, Chimecho is so good in this current meta since those actually good defensive Psychic types left". No, it's still shit. You'll have to adapt your teambuilding to the new meta and find ways to build that don't utilize a defensive Psychic type. In the same way I think it's foolish to claim Ape got better because Sawk and Gurdurr left.