What is the purpose of the bottommost screw holding the IPhone 5 LCD shield plate?
The missing screw from the heat shield in your image shouldn't affect your iPhone 5 in the way you're experiencing. Its only purpose is to keep the heat shield in place, which in turn protects the LCD from excess heat. 

In your case, since the heat shield is still there, the digitiser/LCD shouldn't be affected by any excess heat, nor the possibility of cables moving (as an added bonus, the heat shield does sort of help keep everything in a place).

It's most likely your problems have been low quality replacement digitisers and/or a damaged cable. However, I have personally seen one instance of an iPhone (not an iPhone 5, but that probably doesn't matter) where a cheap 3rd party digitser/LCD replacement only worked properly when the heat shield was removed altogether. 

I've also heard that (but have no personal experience of) some repairers actually removing the heat shield so that their replacement digitsier/LCD screens work properly.

I suppose you could try doing the same as a 'test'.