The black arrow was magical, but there's nothing to suggest that it was an arrow of slaying or anything of the sort, as is often suggested.
Other than every other arrow bouncing off and Bard needing to use that specific arrow? I think it's pretty well inferred it's magical.
"Black arrow! I have saved you to the last. You have never failed me and always I have recovered you. I had you from my father and he from of old. If ever you came from the forges of the true king under the Mountain, go now and speed well!""
Sure seems like a classic powerful magical item to me. That description fits with how artifacts in the DMG were described.
I'd bet the vast majority of stories where a hero defeats a dragon or other such powerful monster, there's a magic item or two involved. Which makes sense, because highly crafted weapons like swords back then were cherished items of their own. We've got a lot of mythology around the magical power of uniquely named weapons:
Areadbhar: spear of Lugh
Ascalon: the sword/spear (it's been used for both) of the aforementioned St. George
Clarent: Mordred's sword, cousin to Excalibur
Excalibur: the most famous of them all
Durendal: Roland's sword
Fragarach: Manannán mac Lir and Lugh Lamfada
Gáe Bulga : spear of Cú Chulainn, given to him by Scáthach
Gram: used by Sigurd to slay a dragon
Gungnir: Odin's spear
Hrunting: Beowulf's sword
Joyeuse: The sword of Charlemagne
Lorg Mór: The greatclub of Dagda
Mjölnir: Thor's hammer
Singing Sword of Conaire Mór
Skofnung: The legendary weapon of King Hrólf Kraki
Sword of Freyr
Tyrfing
Fail-Not: Tristan's bow
You get the point. But there are a lot more than this:
Angrvaðall
Armor of Beowulf
Fail-Not
Mistilteinn
Shoes Of Víðarr
Svefnthorn
Ægishjálmur Helm Of Awe
Bradamante’s Lance
Carnennan
Cohuleen Druith
Dyrnwyn
Eldhrímnir
Gunnar’s Atgeir
Hrunting
Jökulsnautr
Lúin of Celtchar
Mimung
Orna
Peter Stumpp’s Magical Belt
Seven Mile Boots
Singing Sword of Conaire Mór
Stone Of Giramphiel
Ascalon
Chariot Of Morgan Mwynfawr
Dáinsleif
Devil’s Green Coat
Egil’ Wings
Girdle Of Brynhildr
Kantele
Levateinn
Excalibur
Olifant
Ragnar’s Shirt
Svalinn
Svíagris
Swan Cloak
Areadbhar
Armor of Örvar-Oddr
Bragi’s Harp
Claíomh Solais
Clarent
Durendal
Fjaðrhamr
Gae Assail
Gobán Saor’s Axe
Hamper Of Gwyddno Garanhir
Lorg Mór
Mantle of Arthur
Pridwen
Sword Of Freyr
Járngreipr
Joyeuse
Megingjörð
Mjölnir
Baugi
Pair Dadeni
Skíðblaðnir
Skofrung
Tarnhelm
Tyrfing
That all said, I think it's clear to see how the original fighter was mundane with their class being augmented by magical items. The early game emulated a lot of the mythology and folklore, and in mythology and folklore, while heroes did great feats of power without the aid of magic, most epic battles with dragons or other similar creatures did have the help of a magical item. And in 1e, the fighter could use the most magic items.