Any downside other than bigger plug?

And plug is bigger partially because it needs to support 3 phases, which isn’t a thing in USA houses (while common across the whole world) but is a thing at industrial locations. Tesla plug doesn’t work with 3 phases.

Bigger plug is a big one, like you mentioned. 3 phase isn't an issue in the US version IIRC, but it still has more pins than NACS to maintain backwards compatibility.

Thicker + heavier cables too (not sure if that's required by CCS or just an implementation difference w/ NACS). Together that makes CCS much more unwieldy than a gas pump.

The NACS latching mechanism is a lot better- latch failures are a big part of why so many J1772 chargers are broken and unusable.

CCS1 used in North America is only single phase.

CCS2 used in Europe (and elsewhere) supports three phase.

The electricity grid has three phases in the US right? Are devices seldom connected to three phases?

Three phase power is out there but you never see it in residential and rarely in commercial. It's pretty much just an industrial kind of thing.

I have two 3-phase 480V 16A sockets in my 50m2 European apartment.

I disagree with the rarely in commercial. It's extremely common in commercial environments from what I've seen.