Ps2 works fine, gamecube works fine, xbox works fine - what 20yr old console is having issues with an HDMI tv? Every TV I've seen in the past 2 decades includes RCA and RGB connectors.
S-video is gone, and RF is gone. Coax is still there though, so RF modulators should still work too.
Our relatively new Sony TV doesn't have component or composite input, hence the need for a converter. But you want to use a scaler versus a straight signal converter because HDTVs don't always support 480i input (meaning that you wouldn't get a picture at all), and even fewer support 240p, which is necessary for some PS2 and most PS1 games.
A scaler like the RetroTINK can add visual enhancements, but more crucially it ensures that your TV receives the signal in the first place.
S-video is gone, and RF is gone. Coax is still there though, so RF modulators should still work too.