The thickness of the cables is often due to their water cooling. Transferring 1 MW at 99.9% efficiency means 1kW escaping as heat.

That's about the same as a hair dryer. Maybe I can save time by charging my car and setting my 'do simultaneously.

that depends. transferring 1 MW at 1 MV produces very little heat

1 megavolt means huuuge spark gaps, I'd say no less than 20", given dry air. This also means very thick insulation, very bulky and expensive switches and breakers, etc.

Tesla 3MW charger uses 1250V DC: https://en.wikipedia.org/wiki/Megawatt_Charging_System

Liquid-cooled charging cables are a thing. too; one of many: https://www.mouser.com/new/phoenix-contact/phoenix-contact-5...

Yes, but a car can't use electricity at 1 MV and converting the megawatt back to something lower than 1 MV still produces a lot of heat.

Vehicle battery packs are 400-800V today; not 1MV.