Audi is working on a type of electric turbocharger that spools the turbo ahead of the exhaust gas pressure, to eliminate lag . VW is working on one that uses an air compressor to pre-spool the turbo charger. And Eaton has an electric assisted supercharger that helps the supercharger spool, it can also can start the engine, and recover energy from engine braking.

I'm wondering if something like stored kinetic energy like that on the Audi e-tron (a carbon vacuum sealed flywheel) could also be used to ram air into the intake, and how much resistance the air would have against the stored energy.

Well, somewhat, if you mean an electric only compressor. Turbos actually increase efficiency because besides intake compression, they extract energy from the exhaust with the turbine.

Formula 1 cars have a mixture: there's an electric motor / generator in the turbo, so that the turbo can be spun up when there's not enough exhaust yet, but the engine needs a lot of air (when you start accelerating). Or the opposite: when there's excess exhaust energy that you don't need for intake charge anymore (when you've stopped accelerating), you can run the motor as generator providing electricity for batteries. Since the turbo rotation speed is so high and torque small, a relatively small motor-generator will suffice.

Not at all. Problem is that ir has been an uphill battle to develop such technology. The energy required to move a turbo compressor fast enough ti generate boost is very high. A hydraulic approach has also been under works but proved challenging. Although Garret has promoted turbochargers with embedded motors to speed up the compressor speed and reduce lag.

The electric motor to be used would have to be a pretty unusual motor. IMO, supercharging is "better" because the PTO side of an engine is less ... trouble than the exhaust side. Superchargers are generally heavier.

Super chargers generally result in a fairly large parasitic loss to power it. Turbo more or less harvest free energy with only a minimal loss due to increased back pressure.

So they say. The story I always got is that turbo ( for aviation - where this all grew up ) was simply lighter.

The old rule of thumb "turbo costs 3x less than super" is true for a lot of cases. Supercharge provides a linear boost where turbo does not.

Some superchargers provide linear boost, others behave much like a turbo. In any casem FI is used to increase efficiency, so it doesn't make sense to use a super charger with a must larger parasitic loss. Modern turbo can provide boost from 1500 rpm, and result in a flat torque curve - the only negative thing is the lag.