Thank you for comment. Developing such skill would be awesome. I have some grasp of intervals (far from perfect), but not scales, so I have to find the first tone to properly continue. When I find the scale, it's easier to follow the melody, but I can never go error-free on first play real-time.

Do you have any tips about learning this skill efficiently, useful resources, etc.? If I could at least improve in a short time, I would consider investing my time into it.

I understand (a bit) your feel about writing on staff paper. But I was never able to quickly do it and preserve acceptable look of my scores.