Signs of overheated smd chip
I will focus on two specific topics of the original question,


  I am now suspecting manufacturer's fault. Either they managed to burn
  the chips during the reflowing or they somehow used some quick
  test-run lower quality setting on PCB production. I'm just guessing.
  
  So my question is: Are there any telltale signs, that a chip might be
  burnt or damaged during reflow? Would the shininess of chips be any
  good way to tell whether the chip is burnt? Any other possible problem
  causes?


Regarding the first (PCB assembler's fault), it is really difficult to know, however, I would certainly try to unsolder one of the suspicious chips (with a hot air pencil-like gun) and resolder it. It is not as difficult as it may seem, however, you obviously take the risk of not being able to solder back the chip in place.

As per your second question... It is much more difficult that it seems to damage a plastic or ceramic packaged chip during reflow. Unless the assembler's oven is faulty or the operator was really sloppy setting the heating curve.

I would definitely try to clean the residues on top of the cheap, as the shininess looks like flux remains. If the flux is of the water-soluble type, then you have a chance of solving the issue by cleaning thoroughly the top of the IC and also the lateral and other shiny remains on the PCB. This is because water-soluble flux is usually conductive.

How to properly clean the flux? The standard way in the industry is using isopropyl alcohol, commonly known as IPA. It is a good starting point as it is a good general solvent, with good cleaning properties, leaveing no residues behing. Beware of its use, as it is highly flammable and emits vapors. I would also use an anti-static brush.

https://en.wikipedia.org/wiki/Isopropyl_alcohol