How did grunge popularize punk?
Because it is punk. OK, it's punk mixed with glam rock or heavy metal, depending on the band, but it's an outgrowth of the American hardcore scene. Soundgarden was on SST for godsakes, the very heart of American hardcore.
Grunge was just punk rock with the really nasty bits filed off and ready for more mainstream consumption. The Seattle scene had an extreme punk DIY aesthetic, and the music grew out from there, though it also had a heavy dollop of T Rex and Cheap Trick.
This lead to the rebranding of new wave and college rock as "alternative" by the major labels, and they started to co-opt the scene and pump out some of the worst dreck in music history. There's a reason rock n roll is now on life support in the mainstream, and a lot of it dates back to the horrid alt-rock that came out in the late 90s after almost all of the original grunge bands imploded.
This is the normal way of things. Underground movement has some crossover success, but leaves the weirder bands behind. The majors swoop in and sign the more mainstream of the bands and then work to create their own carbon copy knock offs, who almost always suck. They become the new establishment, and then a new underground scene bubbles up. Repeat ad naseum.
Take the Talking Heads, who you dismissed as a "new wave" band. Talking Heads were one of the original American punk bands. They were a vital part of the CBGB scene and were every bit as punk as the Ramones or Television. But as the labels descended and marked some bands (like Blondie and Talking Heads) for stardom, others got left in the dust (like anything Richard Hell touched or the Dead Boys). New wave was a major label construction to cash in on the new punk scene, and Talking Heads fit into that scene comfortably and continued their success. Much like Pearl Jam, 20 years later, would fit in comfortably with the "alternative" scene they spawned. Both bands managed to keep their heads above the general awfulness of the co-opted and defanged movements they largely inspired.
Was that detailed enough for you?