I think that it is a bit of both. You need to respond well to your pro - have a good connection and so on, and in my experience if you see eye to eye, you should be able to immediately appreciate the difference in changes to your swing. Even with my biggest swing change to date, I could see the difference in the lesson that it would improve my golf, but it took 6 months plus to get consistent in bringing it into play in my regular game.
The second thing is that after the last couple of years I am a firm believer in the fact that you can improve your swing your skills and so on and not see an immediate improvement in your scoring. I got to my lowest handicap a couple of years ago now, and I haven't manage to get back there, but I think that in a whole range of areas I am a better golfer now than I was there - I am much more consistent throughout the bag for example and make fewer really big mistakes. So my stableford scores might not have improved overwhelmingly but I would guess that my stroke scores have over the same time period.
Finally, there are many elements in scoring well which are not strictly related to things taught by your pro. Concentration, course management, ability to keep an even keel and not just give up if you start poorly - all of those things will affect your ability to score. So your swing and technique might improve markedly but be prevented from improving your scoring because you are getting in your own way.
I would add health and fitness to that list. If your not in good health or have fitness issues, its unlikely that you will be scoring very well. Its surprising how health and fitness is taken for granted, you don't appreciate it until you lose it.