Can anyone point me in the direction of a legitimate product that will fit my needs?
Looking particularly to relive my PlayStation (original) days etc, a lot of the products I see advertised seem too true to be good...
They are - they're dodgy Chinese products running pirated ROMs... piracy is literally the point of those things.
If you want to relive your PlayStation days then you could buy a Playstation (and a CRT).... but what many of us weren't fully aware of back then was that they crippled the games in the UK market - PAL vs NSTC - our games ran slower (50Hz) and had letterboxing/were squashed - worst case outputting a 576i image in 4:3 but containing a squashed 480i image within it - so you get 17% of the screen taken up with black space where in the US of Japan you'd have the non-squashed 480i image displayed in the proper 4:3 ratio - you could see this in the original Gran Turismo game. (Though unlike the SNES/MegaDrive era they did tend to adjust the speed of play to 50Hz).
Then in some later games they'd try to reduce the squashing effect - Gran Turismo 2 IIRC was rendered at like 512i or something, so you'd get a slightly higher res but also slightly squashed still game with less noticeable letterboxing and adjusted to run at the right speed (but not as smoothly as 60Hz).
This got better with the PS2 - in that case you can output some games at PAL60(480i) - so the proper 60Hz speed and no letterboxing and better (PAL) colours.
So what could you do today? you could get a PS2 second hand and play both PS2 and PS1 games if you like - get an RGB lead or SCART at least... (and/or perhaps look into CRT TVs).
Or you can get the mini-console - the PS One comes with a bunch of games, I've not played it but it should solve the old letterboxing issue as (though the UK version contains a mix of old PAL and NSTC ROMs they're having to scale the image regardless up to fit a 720p output so the image should take up all the vertical space AFAIK - should in theory look good on a 4k TV set as that's just a straight integer upscale but in practice they don't necessarily do that behind the scenes - perhaps someone who owns one can comment). Obvs that does still mean I think there are some games running at 50Hz still but given the image issue is solved and the game speed was adjusted for 50Hz by that era then the only niggle there is some smoothness issue.
tl;dr
If you specifically want to play the original playstation games then I'd probably just get the PS One mini console - legit product and comes with a bunch of games + no faff to connect to modern TV.