In my opinion short game is moot unless/until you can get around just about every green in regulation. Not necessarily on every one, but near almost all of them.
In 2013 at the start of the golf season (march in MD), I was a 15.5 hc and had been playing less than a year. I ended the 2013 season (November) at an 8.3 hc. No lessons and I had been playing for less than a year and a half by the time the handicap season ended. How did I do it? I increased my average GIR to more than 7 per round and I was consistently around almost every GIR. Everyone has a bad hole here and there but I was consistently getting on or around 16 or more greens in regulation per round. Thus my focus for all of 2013 was to improve my ball striking which would in turn yield higher GIR results which it did.
Now, for 2014 that my ball striking was at a decent level (it's still improving btw), my focus was on the short game for this year. Ball striking is still improving but not at the same rate. So now I'm averaging 50% GIR per round. This is great but there was a huge opportunity for me because my scrambling percentage is/was so low. My short game has improved quite a bit but there is a long way to go. I am only at about 33% scrambling but although that seems low, it has made a huge difference for me in getting a few extra pars per round versus when my scrambling percentage was barely double digits.
A great short game can save you from a lot but only if you are around the green in regulation. Last Sunday I had one of the worst ball striking days of the year. I did not feel good and was just "off". I hit only 5 GIR, the lowest total I've hit all year but I was able to shoot 7 over 79 because I got up and down for par 6 out of the 13 GIR I missed. That was the best scrambling % I ever had in a round and showed me that the work on my short game has been paying off.
Like yourself, I was also curious at how I stacked up versus other golfers statistics wise throughout my journey so far. It's hard sometimes because not everyone keeps their stats but there are some good examples out there. From what you wrote it sounds like you are right where you should be based on your scores.