There's a ton of fun options for more raw sounding setups. And there isn't just one punk. Tons of spinoffs and gatekeeping puritanism, just like many other genres. There's a ton of oddballs in the Helix that have turned up on punk, New Wave, pop-punk, or modern punk recordings.
Helix's take on the Rectifier lacks the modes that a lot of rockers and lower gain players use to make them work. Not surprised it was too much. I was surprised to hear Lindsey Buckingham uses an SD-1 into a Rectifier live, but even with his settings I couldn't get anything close out of Helix's Recto model.
Given your tone description, I'd suggest trying a boost [Horizon Drive, Top Secret OD, Prize Drive, or Stupor OD] into one of the following:
Brit 2203, which doess the classic JCM800 sound complete with bright cap, for that 80s kerrang. Line 6 Voltage and Park 75 some easier to dial in Marshall variants.
The MOO))N (Sunn Model T) is fairly similar to a late bass model JMP, IIRC, but pushed it's like a growlier and rattier sounding JCM800 crossed with a Bassman, with an interesting rasp. I like to use the Moo))n Jump, and play with the bright/normal channel drive balance, but you may prefer sticking to the Nrm or Brt model.
The Mandarin Rocker [Orange Rockerverb] is worth a look, for something that does a broad range of rock to metal dirty tones.
Or one of the high-gainers with a Scream 808, Pillars, or other bass cut pedal [even a low-cut EQ]:
For gainier modern sounds, the Panama [5150], PV Vitriol Crunch or Lead [Invective], Placater Dirty, Archetype Lead, Solo Lead Crunch or OD [Soldano], and Line 6 Badonk are all worth a look, too. Or how about the Revv Generator Purple with the least gain & lower master settings you can stand if you want punchy, tight crunch. Purple may not even
If the Revv is too tight/not enough bass and doesn't do it for you without a TS, I'd probably look at the Archetype Lead next, Badonk, or Placater Dirty with the sag turned down.
Oh, speaker cabs are a huge chunk, I like the 4x12 MOO))N T75 for G12T75 sounds (scooped, but punchy), 1x12 Open Cream [G12M-65] is another speaker common in late JMP/JCM800s, or any of the Greenback or Blackback cabs are various flavors of classic, and of course the V30 cabs do their mid-forward thing in variations. Don't stop at the default speaker, try variations! A lot of punks used whatever they had/could afford! Heck, the 2x15 US Dripman makes for a gnarly crunch tone for dark rhythms.
Hope all that gives you some ideas!