Can we build the math from the ground up? (And does Vancian help or hurt that?)

[NigelWalmsley]

What is something wizards can't do that can still be done in D&D?

It seems to me that the onus should be on you to prove that Wizards can do everything that can be done. The Wizard's spell list does not have every spell on it, and there are plenty of things that aren't spells out there. In terms of an answer to your question, I would say that what the Wizard can't do, generally, is compete with a specialist in their specialty without spending build resources, provided that specialist is of a class comparable to Wizard in power in the first place. So a Wizard might be able to out-melee a Fighter or out-skill-monkey a Rogue just by picking the right spells, but they can't out-melee a Druid or out-skill-monkey a Beguiler unless they sacrifice real power for it.

If you're starting from the ground up, and then testing legacy spells against a designed system, that wouldn't be true. Decide what damage a cantrip, 1st level, 2nd level, 3rd level attack spell should do, for single-targets, for AoEs.

I don't really get how this is responsive. Sure, you could normalize everything to one damage progression (you should not actually do that, but you can). But that doesn't change the fact that Burning Hands is a melee-range AoE, Scorching Ray is a close-range single target attack, Fireball is a long-range AoE, and Wall of Fire is a medium-range BFC effect that's not even instantaneous. How are you going to set things up so those are all modes of the same underlying "Fire" spell in a way that isn't more trouble than just having four different spells?

(I'm not going higher than that for daily spells in my system. It's either just bigger numbers on a treadmill, or effects that obsolete martials.)

I'm curious how you think Dimension Door is either numbers on a treadmill, or able to obsolete Thor.

I think the thing to do is set up a paradigm where "buff-spell alone << expert ability << buffed ability". Which would nerf (or nuke) a lot of low-level skill-obsoleting spells--looking at you, jump and spider climb.

I think maybe you intended this to reply to something else? I don't see how it's really responsive to what you quoted.

Balancing things based off different levels of dependency on resources doesn't generally work, as each table is going to have different numbers of enemies per encounter per day (unless you can find a way to make those numbers rigid).

I think it works well within the context of a specific encounter. If you set things up so that the standard encounter has 5 enemies and lasts 3 rounds (or whatever numbers), variable resources within that encounter give DMs a powerful tool to fine-tune balance during their game. If the Warlock (who gets at-will Invocations) is underperforming, have a longer fight. If the Wizard (who gets a bunch of AoE effects) is underperforming, have fights with larger numbers of enemies. And so on and so forth. Even if your game is mechanically perfectly balanced, it won't be balanced in practice, so you need to give people tools to adjust in practice. In this case, it also simply makes for more interesting gameplay.

Combat isn't the only metric, unless it's the only way to solve your problems. Treat Combat and Non-Combat as two separate modes of play, and make it so you only ever have to compare Apples-to-Apples and Oranges-to-Oranges.

There's a limit to which this is possible. Many abilities aren't clearly divisible into "combat" and "non-combat". A classic example would be Silent Image. You can use it in combat to trick enemies, but you can also use it to avoid combat entirely, or in social situations. I agree with your general point that you cannot and should not be balancing characters as having a total number of points to stick into a combination of "combat stuff" and "non-combat stuff", but neither can you treat them as entirely separate.