On the other hand, kids that young just don't think about organising stuff like that. They just take each moment as it comes and it's great for a mom to encourage friendships.
We had a kid my son's age who lived just around the corner for years. You'd think that they would be playing together in the street and best friends since they were even in the same class. Nope, wouldn't have anything to do with each other. Years later, after the kid has moved, they are good friends.
I dunno - my kid used to ask me if her best buddy could come over, back when she was in nursery school. Unfortunately, we lived in a neighborhood that made that impossible, but we did make arrangments for the two of them to play at a park in a better area every couple of weeks, with one or the other of us parents doing the picnic lunch. But the Daughter is an odd bird (she was a bit, no a lot, on the precocious side) so ....? I do think it's better for the kid to ask for things to be arranged, as just living in the same street/of the same age/at the same school doesn't necessarily mean the kids like each other or want to hang out.
Stumbling down memory lane here - I threw the Daughter the local equivalent of a Chucky Cheese birthday party once. (I knew the manager and we got a huge discount - basically all I had to pay for was the pizza at cost.) This was, I think, kindergarten. Sent invites to the class of 20. Got a handful of RSVPs (the rest didn't bother to reply one way or the other). Only one kid showed up on the day (we were expecting six) and her babysitter also came. So she had a good time anyway.
I later made the mistake (4th grade, in another state and a nicer neighborhood) of throwing her a surprise birthday party and inviting several of her classmates. Turned out that a couple of them were ~not~ her friends and it was awkward. Once those girls went home, and the sleepover began, it improved a lot. But ever since then I've been in the "hands off unless asked" mode regarding her social life, not wanting to be too helicopter-mom-ish. All that kind of makes me look askance at parents "arranging" their kids social lives - unless asked to do so. Which is another thing entirely.