RCF, I totally agree with your description that the chick looks like it is "swimming on the floor of the box". I mentioned that I’d like to see more gravel in this box and cannot understand why someone who oversees this site has not attempted to correct the problem by now. They must be watching - so it doesn’t look hopeful that they will.
The problem might not be noticed if watching briefly, but if you look at the size of the egg (which is about 2 x 1.6 - 1.9 inches, I read), and compare the egg to the open floor area of their scrape (which is usually about 2 x 9 inches), then also consider that only a few pieces of pea gravel are usually seen on this open floor - and you might agree. I read that falcon nest boxes require an average of 4 - 6 inches of pea gravel, and from what I can see I have to wonder if there is even one full inch in that box. Adult falcons have long talons which should not be flat on the floor of their nest while sitting on eggs or chicks, and the chick really needs something to grip! The floor seems to appear extra slippery at times. I certainly hope it doesn’t actually have an extra smooth surface! (Maybe it just seems that way since the chick tries so hard to get itself up.) Otherwise, the new box looks great, and if this pair did have a successful clutch in the older box (as the article I posted mentions), then perhaps it was prepared in a better way. I’ve heard of falcons nesting successfully on rooftops or gutters, but maybe there was a little something more for a chick to grip - such as dirt and weeds, though I’m just guessing.
There have been quite a few flies in the nest as well, though flies are constantly attracted to these nests (and normally without problems). However I'm also aware of very serious problems that certain flies are able to cause in birds (and other animals). Some of these problems tend to occur if the animal is on the ground, fairly still, or happens to have an open wound (even if the wound is very small).
This particular site is accessible, but I imagine it's too late for gravel alone to help without treatment. Besides there might be another type of health condition causing the weakness, though so far chick is still vocal and eating well. Whatever the cause, I still feel the lack of gravel (on what appears to be a very slippery surface) has contributed to a noticeably uncomfortable and stressful situation for parents and chick.
Hopefully someone with more insight will add their opinion, as well. This little chick is weakening further from so much struggling, I believe.