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(?)