What you've described from your own experience is the substance of what a professional network is, and has always been. LinkedIn connection requests are trying to create a digital product in the form of the real-world phenomenon that you've experienced and described. If we're going to call one of them a professional network, everyone in this subthread is saying, let's give the real thing the name professional network, and the imitation a different name.
(None of this is intended to disparage LinkedIn. It is what it is, but if it's trying to be a substitute for real relationships between humans, it will always be the shadow, not the substance.)
I don't think the type of network I've just described is imitating anything. They are mutually beneficial but purely professional relationships. The fact that the internet has enabled people to have a much larger number of superficial relationships doesn't mean that what the poster was describing is the only way to have a "real professional network". Yeah it's a professional network, but it's not the only kind.
It's a tough one, because while I can see that it is also a professional network, said network would be pretty offended if I called them that. So I don't necessarily disagree with you, or even with the other commenters, but there is at least some room for subjectivity about what to call your own personal relationships.