> It runs but can't connect to modern Wi-Fi without an update, which it can't access without an update to its firmware (a classic catch-22).

In this case the issue is that PSP is so old that its hardware simply cannot connect to WPA2 networks (it can connect to WPA and WPA/WPA2 Mixed networks however). This is not something that a software update can fix.

Also, honestly, internet access is honestly not very useful for PSP, especially now. Pretty much all games have shutdown their online services, games themselves can be played from Universal Media Disc, and the web browser is pretty much useless with modern web, and with 32MB/64MB of RAM that the console has I don't think Sony could update web browser even if they wanted to.