Incidentally also why I'm not a fan of weapon specialization. If the fighter finds a weapon, I want him to be able to pick it up and use it effectively whether it's an axe, a sword or a bec-de-corbin.
Not a fan. Gage Infernus, the Great Harvest, whose prowess with his horrifying scythe is the heart of his legend... should not be reduced to using a dopey cavalry saber because the dice said so.
I'm of two minds about this. I'd very much LIKE characters to be able to use and be (if the player chooses) in part defined by their signature weapon, but I also want a Fighter to be able to competently use whatever they find, and not be hobbled when they're deprived of their signature weapon.
In my current Old School 5TD game there's no specialization, and the best Warriors we've seen have used the best weapons they've found. The highest level one currently, Graves, has spent most of the campaign wielding an arming sword and shield, mostly wanting the protection of that shield. For quite a while he used the Axe of the Vexx as his main weapon, a +1 Waraxe which could be commanded to release a cloud of poisonous fumes around the bearer; potentially very deadly against a horde of foes, but tough to use when friends were nearby. It was shattered in a dungeon being used to hew apart a magical statue, and now he bears the Sword of Sleep, an arming sword +1/+3 vs Undead, which glows and illuminates a 15' radius when unsheathed and held with blade exposed, but brightens to 30' radius when in Line of Sight of an undead creature. As well as the Shield of Charging, which grants him an additional shield bash attack whenever he's got at least 10' of run-up to engage a foe.
Some months ago, however, he defeated an orc chieftan and claimed as prize the Chaos Blade, a terrible two handed sword +2 that strikes for 2d8 base damage in addition, and can Sweep attack multiple adjacent foes with each swing. Due to its awesome offensive power he's had a meaningful choice in many battles since whether to maximize his AC or to go to town with the Chaos Blade.
If one of the fighters in my game had a signature weapon I'd definitely be open to him questing for a legendary magical one, or commissioning a master smith or wizard to make him a custom magical version. But there's definitely a charm and joy to fighters being able to find random awesome weapons and make full use of them without a functional penalty because they're not getting their Specialization.