Yup. Further, the charging infrastructure needs to be upgraded if more charge is demanded in shorter time. More current flow means more copper needed in the conductors all the way back to the source grid or local power station.
A Tesla charge point designed to meet only Tesla demands will only deliver Tesla recharge time, so the benefits of this would not be experienced with the current infrastructure.
This is part of why eventually, after the spell of EVs is broken, we will inevitably move to fuelling infrastructure more like fossil fuels, but it will be hydrogen produced mostly by domestic and community solar that is the fuel transported and all EVs and ICE vehicles will be converted to run on hydrogen, imho.