The big thing everyone's glossing over is that neem oil doesn't kill the eggs, so you're in a cycle where you spray, kill adults, then the eggs hatch a few days later and you're back where you started. That's probably why two applications barely dented it. You need to spray every 3-4 days for at least 2-3 weeks to actually break the life cycle, hitting those undersides hard each time. Are you doing that, or were you spacing them further apart?
Beyond the heat and drought angle (which yeah, those definitely make mites happier), consider whether you changed anything else this year - like mulch type, nearby plants, or even if you brought in new stock or borrowed tools from someone else's garden. Mites hitch rides on everything. Also, even if your area wasn't unusually dry overall, sometimes a patch of your garden ends up more stressed than others depending on sun exposure or soil drainage.
Before you rip them out, I'd say grab a miticide that actually targets mites instead of relying on neem - something with sulfur or a proper acaricide if you want to go that route. Dahlias are worth saving if you can knock this back. The plants themselves are probably fine underneath; it's just the pest pressure that's new. What's your spray schedule been looking like, and are you getting good coverage underneath the leaves when you apply?