I think re the tutorial there was a missing instruction about wall jumping. I eventually figured it out on level 1 and it made me think back to where I got stuck in the tutorial and I realized it was that mechanic which would have progressed me.
I definitely want to reiterate that you nailed the retro vibe! It made me feel like I was trying out shareware Turrican on my Atari ST again - eg, I played the original versions of the games you're winking at and your nod hit the mark. It definitely felt like those old games!