that would involve unannounced channels too, Lightning network channels are announced to other nodes by default. Sometimes unannounced channels enable types of attack, but at least in the case of spamming, they help mitigate against attacks.

With good routing algorithms (i.e. involving multipath routes) and effective anti-spam, I would imagine there would fewer benefits to unannounced channels. But this of course means that LN nodes with a complete network-wide routing table still know the channel balances, and so privacy is harmed. IIUC, it's not possible to run the LN effectively if all channels are unannounced, and realistically it's impossible to have entirely private channels (as your channel counterparty always knows the channel exists)

this does not preclude the possibility of having 2 or more Lightning nodes, of course (I would suggest that most people running their own nodes will end up opening a 2nd, if only temporarily to improve anonymization of funds).