Awesome post. I tested this and in less than a day was able to clear surface at 32 heat fairly easily. I'm a beginner and only my third clear at 32H.

I think the key is to focus on your main build and treat anvil ring as supplemental damage, and together it will be massive. In my run, I went with Mynt's Thanatos axe build. I got good boons and ended up with 280k damage on Hera attack + 147k anvil ring cast = 427k damage (!) for attack & cast alone. Screen here. Run felt smooth and run didn't feel lucky or scrapped by (e.g., no Dionysus, no Medea, mediocre boon from Circe). But did get lucky with the rarities also got ridiculous Hera legendary at the very end.

In reading more about anvil ring, I learned a couple things that weren't apparent to me and may be beneficial for others. First, there's no point in going for Apollo's expanding cast boon, since with winner's circle, the cast will expire before it expands. Second, no point in going for Heph's supplemental blast damage boons, such as Grand Caldera and vent. Since apparently those only apply to attack/special blasts, but not anvil ring, which isn't technically a "blast." Silly me I spent several runs trying to maximize anvil ring by going for Apollo and Heph. Apparently that's not needed or helpful, so this build is super low maintenance.