It's partly because in 8-bit computers the programming environment was the first thing you were exposed to. You weren't force to program though, you could limit yourself to LOAD "*",8 - but also you were free to experiment with more commands if you wanted to.
In newer computers, the programming interface was replaced by something else, and then completely removed from the system.
Is your Play Station 2 PAL (UK) or NTSC (American)? The Spectrum +2 is (I would assume) PAL.
I'm reaching, but if your PS2 is American and thus already NTSC and works, then your converter may be expecting an NTSC input. If you convert the PAL to NTSC before feeding it into the HDMI converter, it should in theory work.
Think that analogue signals have a lot of tolerance. The Spectrum PAL signal was very far from followiing 100% the PAL standard.
I would assume nothing needed to be exact because you could fine tune on the fly. I remember when you tuned our 3 TV stations in the UK by dialing the frequency dial to the correct radio frequency. It's been a long time that I don't recall the radio frequencies by perhaps I recall ITV being 68 MHz(?)
It took less than 30 mins for my 11-year-old daughter to figure out how to draw basic rectangles and circles in various colors on the screen. She was hooked. And I was transported back to my childhood. I saw the same joy I experienced as she discovered new ways to have the computer do her bidding.
Today’s computers can do amazing things, but they sort of miss out on the whole human user experience bit. Maybe they need to go back to basic ;-)