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.)