If people had a pipe delivering petrol or diseasel to their homes so they could top their tank up every night how many would need to refill elsewhere, and how often?
Well I can have a stab at that, having been in an analogous situation. There's an old joke about Rover V8 powered Land Rovers - they'll pass anything but a petrol station

One answer to the cost of feeding such a beast is to run them on propane, a.k.a. LPG. When I first converted my old Disco, LPG was available locally but it was rather pricey - and I could buy 47kg cylinders and fill up for considerably less, even after paying fuel duty etc. So I was mostly filling at home, and that carried on when the Disco got replaced with the 110.
Running a thirsty beast on LPG had a lot of similarities with the discussion of EVs. Range is limited (by gas tank capacity) and filling points aren't anywhere near as ubiquitous as for petrol/diesel. I used to subscribe to an outfit that provided maps of LPG stations, and later provided a file to load into the GPS.
Familiar routes were/are fairly easy - you get to know where the filling stations are and when they are open. But other trips involve a bit of forward planning. As an example, when my mate would be doing rallies in Wales, getting gas at the weekend was often a problem and we usually took our own. In many cases the map would show plenty of places to fill up, but they wouldn't be closed at the weekend
But in answer to the question. For my daily commute I could fill at home. For some trips I could manage but range anxiety could kick in - especially when the little used petrol system started playing up and removing that backup option

Longer trips required either taking our own gas bottles (can't do that with a pure EV), or careful planning to use whatever infrastructure was available. As to what proportion of usage anyone could do from "home filling", that will vary considerably between users. Back then, I'd guess that perhaps 3/4 of my use could be home fill - these days probably a bit more, but I would absolutely need public facilities.
...coincidentally a period where people will be more inclined to use their car instead of walking/cycling/waiting for a bus?
Not only that, but in winter the lecky requirements will be somewhat higher. Lights, heater, window demisting, will all take their toll and significantly reduce range/increase charging requirements. At the same time, this is the time of year when lecky demand is highest.
It's almost always blowing somewhere, day and night.
So wind can "almost always" keep the lights on

But actually "almost always" is rather over-stating things a bit - there are quite significant periods when there's insignificant wind across wide areas. I can't remember where I read it now (probably one of the IET journals), but someone did a study and correlated data across Europe and showed significant correlation between calm spells in northern and sourthern Europe - in effect, don't assume that when the wind isn't blowing here, that we can import extra from Europe as they may be in calm as well.
Going back a few more years, I recall reading in one of the IEE journals where someone had studied weather records from a cold and calm spell. For at least 10 days, there was a static high pressure zone over most of Europe and Scandinavia - with a result that there would be little wind power generated across the continent for a significant period. You can absolutely forget about "grid scale storage covering that sort of shortfall.
My question stands - what do we do when there is no wind and no solar?
This is where the "non-engineers" will do some hand-waving and utter things like "always blowing somewhere" and "just need some storage". The reality as as many of us here understand - you either turn off the lights (the primary function of "smart" meters), or you fire up some backup generation. Of course, what the supporters of wind casually ignore when comparing the headline costs of wind and nuclear is that (in general) nuclear doesn't need nearly 100% of "something else" as well. The cost of keeping all those old coal and gas plants available (but not being used very often) is a hidden cost that needs adding to the cost of those intermittent generators.
We could also invest in a solar scheme in one of the sunniest parts of the world. Saharan Africa, perhaps.
And use the generated lecky to make methanol to replace petrol/diesel. We already have the infrastructure in place to store/transport/retail that
Just spoted
this article on BBC news, somewhat relevant to current debate. Particularly the question of if we have enough electricity.
And repeats the usual misdirection that we have a lot (and growing) of low/zero carbon lecky now. That's irrelevant because until we get to the point of having such low/zero carbon generating capacity spare, then adding load to the grid will open the taps on a fossil fuelled power station somewhere. So until we do get to that state, then lecky cars are absolutely not low/zero carbon.
With a great many countries interconected it would have to be quite an epic fallout with multiple countries to pose problems, and would be self defeating excercise for all sides.
Of course, we've absolutely no experience in the last century or two of fallings out between countries - and things look oh so stable right now as well

It probably wouldn't take too much of a falling out to destabilise the network - and it could even be a tactic of war to actively do that. Such a network would certainly be a strategic target.