Chris Paul is a better more impactful player than anyone on those Hawks teams, but Chris Paul is also not a top 5 can't miss type player.
I know we have a thread comparing the C's to the 04 Pistons, but it is really the Suns. Like Chris Paul, Ben Wallace had a tremendous impact to his team's success. Obviously not the scorer of Paul, but Big Ben was such a force defensively and on the glass, he could alter a game. The Pistons had some excellent all around players like Billups, Hamilton, and Sheed (similar to Booker, Bridges, Ayton) and then a lot of depth i.e. Prince, Okur, Willimson, Hunter, and then vets like Campbell, James, and Ham which I'd equate fairly well (at least from a talent standpoint) to Johnson, Payne, Crowder, Shamet and then vets like McGee, Craig, Biyombo.
The Pistons went from out of the playoffs in 01 to ECS, ECF, and then winning in 04, then back to the finals in 05, and then 3 straight years of ECF. Their championship is considered a bit of a fluke, but they did sustain success for a very long time (6 years of ECF or better). The Suns obviously are newer in their run, but the Pistons got that championship in year 3 of their real run, so if Phoenix can stick around awhile, I think that is the comp. Of course, Chris Paul is getting super old so this might be their last shot, especially with the Ayton uncertainty.