If “near future” is 3-5 years then yes. Four interesting protocols in early development: Optimism, Arbitrum, zkSync, Scroll. These already all compete with PayPal in fees & transaction time, but not in UX, features, and widespread use. In theory they could provide similar frontend web UX, but with the option to hold tokens non-custodially so you could withdraw & exit the protocol without having to ask permission.
(IMHO it’s likely that other tech/protocols will emerge in 3-5 yrs that supersedes the protocols in development today.)
Lots of typical crypto caveats - eg: USDC on an L2 is centralized around Circle’s ability to redeem. Protocols can have bugs that make coins go poof. Non custodial ownership is harder for many users than asking PayPal or a bank not to lose their funds. L2s specifically are typically run by a single sequencer who could potentially disrupt your ability to use the network smoothly (but then you could use an escape hatch to permissionlessly get your funds out if that happens).