>Third-parties take care of this bureaucracy for you. Check out the Phoenix wallet, for example.

Sort of defeats the purpose then, doesn't it? Why not just make lightning payments directly from your bitcoin exchange?

In the case of Phoenix, you have semi-custody - i.e. control over your coin, and it can't be taken from you. There are other non-custodial wallets (Wallet of Satoshi) which are like an exchange - so they can rug pull you anytime just as Paypal can.