Yes, the grid is "big enough" and (hopefully) "smart enough" to compensate peaks, the possible issue I was pointing out was not at "grid level", but rather at "local neughborhood" distribution.
If the transformer serving a given number of houses is (say) 300 kVA, it remains 300 kVA, and as well the actual distribution cables/wires are rated for (still say) 100 kW each, they remain that size/gauge and both will likely need to be replaced by "beefier" stuff.
BTW I took the US as an example because they - on average - consume already a very high amount of electricity pro-capita something like 12,000 kWh per year:
https://en.wikipedia.org/wiki/List_of_countries_by_electrici...
and so it has surely an already well dimensioned infrastructure, but besides the nordic countries (Norway/Sweden/Finland) Europe is less than half than that, Germany and France around 6,500 most of the other EU countries around or below 5,000.
Besides there can be some repercussions on single homes as well, depending of course on local Laws, to give you an example here in Italy there is a "limit" at 6 kW, for constructions until 2008 there was no need of a project for the home cabling for power consumption below 6 kW while since 2008 the project is always compulsory, and 6 kW it needs to be made by an electrical engineer.
Since most houses are < 6kW, very few, even of the recently built ones have a project redacted by an electrical engineer and the moment you ask the provider to get (say) 10 kW you will need this project which will cost you (only the project) something in the 1,000-1,500 Euro range to which you have to sum the actual electrician/contractor work to add a line for the charger, probably another 1,000-1,500 Euro.