Depends. A single battery cell would have nontrivial resistance, yes.
But a big bank of batteries, like are in an EV? Very hard to give them enough current to heat them up. Most of the "heat problem" is from the bottlenecked current path into the car; once you fan out across all the individual cells, each individual cell isn't receiving much current.
And a bank of supercapacitors? You could charge it effectively instantly.
The current is limited by what the battery chemistry can take, not by the cables. This is why the first 80% can be charged quite fast in modern EVs, and the last 20% are really slow.
Additionally you need to have the current to deliver in the first place. Having a grid that can dump 25-100 kwh into any given car in a couple of minutes is no small task if everyone is doing it.
The utilization factor would obviously be much lower than it would be if everybody charges at a lower rate so if the total amount of energy is equal that just means that individual vehicles will spend less time charging, and the grid will see - roughly - identical utilization on average but the peaks may be higher.
Probably worth pointing out that the peaks and troughs are what are challenging to deal with. Generators aren't generally great at changing output super fast.
I keep hearing battery tech is getting good, and the research I've seen suggests that more storage on the grid would improve efficiency by a lot, so I don't know if it would even pose a particular challenge if that sort of demand arose.. but overall utilization isn't really the limiting reagent.
But a big bank of batteries, like are in an EV? Very hard to give them enough current to heat them up. Most of the "heat problem" is from the bottlenecked current path into the car; once you fan out across all the individual cells, each individual cell isn't receiving much current.
And a bank of supercapacitors? You could charge it effectively instantly.