It appears to me that carbonating a cornie keg laying on its side should make the carbonation go faster as there is a larger surface area between the beer and the headspace. Could laying the keg on its side have some kind of down side? Would the petroleum grease I’m using to seal the main hatch absorb flavors? Thanks
My CO2 is outside the fridge. I have two regulators. One goes to the kegs inside, one has a long hose outside. To force carbonate I crank that puppy up to 30 psi, and then put a bar stool in front of the fridge, put the keg across my legs and gently rock it back and forth for about 7 or 8 minutes. You can hear the CO2 going in. Then I hook it up to service pressure.
It works really well, the beer pours perfectly in a few hours. can see no negative effects…
Assuming it is food grade and made for kegs like this product it should be fine. Laying it on its side will also be fine, and may or may not be faster. For me, I’d have to do it on the floor so it would be warmer and annoying. 
Yes I’m using petrogel. I may be one of the few people who is kegging without a fridge or freezer relying on the coldness of the basement in the winter. So I might be one of the few people for whom laying a keg on its side overnight is a good option for carbonating.
In addition to increasing the surface area by lying the corny on its side, you can also try to get the temp lower (but not too low–i.e., freezing) to speed up the CO2 gas going into your beer.
Most of us just set it and come back in a week.
I would agree with just setting it and coming back in a week. The keg acts like a bright tank if you let the beer sit there in cool temps it drops most of the yeast and clears the beer. I suppose if you were in a crunch you could do like the OP said, but I’ve learned that patience is key to good beer!