Those talking down flaming arrows because they aren't useful against goblin raiders need to expand their paradigm beyond the current sieges. Someday, we will get to siege other people, and we'll want flaming arrows.
Imagine using flaming arrows against people in dwarven wooden armor, fighting to defend and elven tree city. Even assuming that the elven AI is as nonplussed by being on fire as the dwarven one, in this hypothetical time, a flammable fort full of flaming elven tresses and robes and armor and swords is going to become a massive collateral damage issue. A small party of torchmen or fire archers could devistate their population and economy for decades to come.
Hairy, hostile titans might also be good victims for such a weapon.
Again, adding fire to a melee weapon that isn't designed for that (like a torch, that burns way out at the end) isn't a good idea. Even then, if the non-magical flames are hot enough to do any real harm on flame contact, then they're going to slow roast the user like a pig on a spit with radiant heat.
Aside from managing the fuel, safety and heat exhaustion issues, you don't really want to be setting fire to things that are going to grab on to you the next second, or buildings that you are inside.
i"m all for weapon upgrades, but fire is just really impractical to keep that close to you.