Should Sony also be forced to keep their service running for PS1 games that had an online component - forever?
On the other hand, would that even be particularly difficult?
For most companies making games with an online component, if they set aside 0.1% of revenue into a fund that would take over server costs once support ends, that fund would last for a very long time.
At a certain point the player base drops enough that you only need one server and a few hours a month of labor. (Less than one server, actually, once you factor in virtualization. Even if you want to have some redundancy and support multiple regions.)
As for Sony, PS1 games and on-line: similarly, whatever middleware there is that facilitates on-line gaming, endpoints should be user-configurable. Fortunately, at least PS1 the device doesn't require Internet to run.
(Come to think of it, did PS1 even support an Internet connection?)
Yes! Absolutely they should.