I must confess, I'm not exactly sure what we're voting for in this scenario!
OP, there is nothing wrong with your colleague organising after work drinks every so often. There is also nothing wrong with you not going because you have other commitments.
You speak about why you can't go, but you also point out that you don't want to because you're tired because you're a working parent who doesn't drive... but however they rearrange drinks, you're always going to be tired in the evenings?
So your colleague isn't unreasonable to organise drinks when they do, because you're never going to go because you don't want to participate. You're not being excluded.
I think it would be nice to make the effort and occasionally go. With enough notice, you could arrange one-off cover for that pickup. You could. You just don't want to.
And re EDI compatibility - I think it's important to suggest drinks on different days occasionally (if you're always organising drinks on one colleague's non-working day, that could be seen as exclusionary). It's also nice to suggest the occasional team lunch, as that's within working hours, and thus widens up participation to people who cannot stay late after work.
However... I suspect the OP wouldn't want to go to a lunch either.
It's not a criticism - some people don't like socialising with their colleagues. As long as they get on with people at work during work hours, that's fine. But I don't know why it's so hard to just admit outright they don't want to do that, rather than inventing many obstacles as to why they can't.
"I don't want to" is an acceptable reason not to do something optional.