His alleged military badassery is sometimes used to handwave his epic CQC. The idea is, dude believes he's a beast, probably saw some action movies as a kid, poof, instant badass (within the context of the magical Otherworld). If this holds up, then the same could be applied to his marksmanship.
The limited weapons of Downpour might have been a response to "Origins", which featured Travis packing up to six guns, plus assorted knives, scalpels, cleavers, katanas, sledgehammers, billy clubs, toasters, televisions, not to mention ammo and healing items. Travis would have needed his truck to haul around all that stuff.
I agree, I didn't like the way Downpour's weapons were so fragile. Some items, like chairs or cheap cutlery, should have broken quickly. But a pickaxe (a took designed for puncturing rock as opposed to soft squishy monsters) should have been sturdy enough to see him through a good part of the game.