One of the advantages of the first materials you cite are lightness.
Tungsten is a very dense and hard material, and would contribute to the weight of a firearm without much benefit. It is also very difficult to form and shape*. I do not know specifically, but would not expect tungsten to be used in firearms themselves. I am "correctable" on this point, however.
Powdered or granulated tungsten was used in certain experimental lead-free bullets to contribute to the weight (sectional density) of the projectiles, which were basically various other metals like zinc and tin, with the tungsten particles just "mixed in," as opposed to being alloyed with the zinc or tin (or other materials --including plastics).
The tunsten particles were kept "inside" the bullets so as to not damage the barrel's rifling.
There were also tungsten cores for many armor-piercing bullets.
Terry, 230RN
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* The process for making light bulb filaments has been pretty much nailed down, but it involves compressing powdered tungten and "sintering" it, where the resulting mass is heated almost to the point of melting so the particles more or less fuse together on their surfaces. From that point on it is difficult but doable to draw the thin wire from which the filaments are made.