The fundamental part of Lightning is the protocol used between any two peers - trustless, out-of-band transactions, with delayed (possibly indefinitely) settlement.

Two peers with an open channel is all there ever needs to exist.

The "network" part just allows these pairs to form a connected graph, allowing routing and therefore payments between otherwise unconnected nodes.

What the question describes is essentialy what how the network would operate if, say, ACINQ and Phoenix nodes on people's phones was the only thing in existence, forming a singular hub-and-spoke architecture. Phoenix nodes ("clients") are restricted to form channels only with ACINQ's central node ("server").