What has old Potassium metabisulfite turned into?
According to my research, I found the reaction of oxidation of potassium metabisulfite but the products of this reaction does not seem to match with your experimental observations.


  The jar has been standing with a a loose fitting lid for thirty years in a cool and dry place.


This means that your old  potassium metabisulfite has been subjected to atmospheric oxidation in a cool and dry place and thus the temperature of the surrounding did not exceed the room temperature.

$$\ce{2K2S2O5 + O2 -&gt;[298K] 2K2SO4 + 2SO2}$$

(source)


  The jar contains a dry substance now like matte flakes. The substance does not react visibly to hydrochloric acid and it does not smell of sulfur dioxide.


Yes, potassium sulfate has a flaky, crystalline appearance. Potassium sulfate does not react with hydrochloric acid. But one of the product is sulfur dioxide and the flakes smell of sulfur dioxide.


  The substance does not dissolve visibly in cold or boiling water. A
  few flakes poured in water remain on the bottom seemingly unaltered.


Potassium sulfate is soluble in water: 120 g/L (25 °C). 


  When heated to red heat in a test tube, it does not melt, the coarse
  powder is not sintered. No water is given off. Some sulfur is
  sublimated, a yellow deposit in the cold part of the test tube and a
  smell of sulfur. Not much sulfur is given off, the volume of the
  original substance is not visibly altered.


Potassium sulfate does not decompose even at high temperatures. But if the surrounding has carbon monoxide, it gets reduced to potassium sulfide which has a yellow-brown appearance when impure.

Summarizing as a table,