Why does solder turn dull grey on my iron and exhibit poor wetting?

Soldering iron tips should never be "clean" from everything. You must maintain a small amount of solder on them at all times to avoid oxidizing the iron plating (as in Fe iron, major component of steel, iron). After brushing excess solder and flux from the tip with a sponge, reapply a tiny bit of clean solder.

If your solder turns a dull grey, that would be due to it's oxidization, which means you're either waiting way too long or the iron is way too hot.

Tip tinner should only be used sparingly as it is fairly aggressive and will ultimately eat through the iron coating.

You get what you pay for with soldering equipment, you can't expect much for $9.95! Get a good quality temperature-controlled iron.

Either it's oxidized, not getting hot enough, or both. Unscrew the tip and check the contact with the heater. Scrub the heck out of the tip with your brass sponge and dip it in a bit of flux. Hopefully it's not pitted.