sounds like the Analog to digital converter went.
it's a small chip that converts the analog signal into something the digital encoder on the TV would understand.
if it were simply the ports or some part of the circuit you'd get fuzzy/distorted signal due to the nature of analog being... well analog.
but if there's no signal going from the A/D chip to the Digital signal processor then there's no signal at all.
this can be bypassed however you are right in that your best bet is to get an external A/D Device.
or you could bypass the Analog inputs entirely with an OSSC based mod and get better than the original signal lol.
as a side note, they make HDMI cables for PS2 these days so that's also an option
they're basically taking the PS2's Component signal, up-scaling them from 240p/480p/480i to 720p/1080p digital signal and feeding them to the HDMI input on the TV.
it's not perfect but they work decently little to no lag from the scaling/conversion process.