I think BaS 2 explains pretty well why this is the case (Spoilers below)

We make it possible for Atlas to attack Rapture and even though we don’t see Rapture fall directly we know the time in which this is happening. Assuming Elizabeth is only out for a few hours or at max one day due to the chloroform and then two weeks because of the Truth Serum (as Atlas says) before she makes it possible for Jack to board the plane, Jack should arrive at Rapture around 3 weeks after the New Years Eve.
This is of course only true if

• I am right about the time Elizabeth is out because of the Chloroform

• Atlas didn’t lie about the time she was out due to the Truth Serum

• Atlas sent Jack immediately on his way topside and then back to Rapture

• Jack’s way out of and back to Rapture didn’t take more than one week

If all of this is the case Jack arrives about 3 weeks after the Civil war started. This is a lot less time than I originally assumed had passed between New Year’s Eve and the start of BioShock 1. Considering how run-down Rapture is when we arrive I always thought at least half a Year had passed since Atlas’ attack.
But the short time would absolutely explain why there are no skeletons only dead bodies.
There were of course victims even before New Years Eve but back then they still took care of the bodies I would assume.
This is a lot of guessing on my side so I can’t prove anything and the writer of the comment back then didn’t have BaS 2 so he couldn’t have known that.
This doesn’t explain why there are no skeletons in BioShock 2 though!