I sort of did something similar to this in my last Dragonlance 5e game, and will likely do something similar when I start it up again. I gave each player a scaling magic item related to their character, most were based off of preexisting magic items from either 5e or another previous edition. Hence why I say 'sort of'.
My players use of them were mixed, particularly as half of them were the sort to never remember how to use them properly (remembering to track resources or spell / day abilities from the weapons, etc.), though the one consensus I had was that even with forgetful players, many of the items were borderline game brrakingly OP. ...though this is coming from the opinion of a DM who likes grounded narratives and feels that something on Par with a Holy Avenger or Whelm, or an Elf Blade is the strongest a magic item should *ever* be. I pretty much straight up toss out the whole benefits/detriments part of artifacts if I choose to include them in my game, or at most allow a player to roll once on a benefit chart for an equivalent detriment.
...that all said, suggestions for how I'd handLe scaling magic items (which are effectively about what weapons of legacy are):
1) All should require attunement. Be it a quest, ritual, or something befitting the item, and they should probably function as a lesser version of the item in the hands of another, if at all. For example, most of my weapons were just magic or a +1 bonus in the hands of someone who isn't the intended character, and would be more powerful in said characters hands.
2) Unless you intend for a high powered game, no enhancement bonus until around level 4 at minimum. Assuming you even wish to give them the weapon before level 4, it should just be treated as a magic weapon. Scaling based on ability score adjustment levels (4, 8, 12, etc.) is likely a good spot to bump up the weapon, be it an enhancement increase, or some other effect, like adding an extra damage die that's elemental or something, or making the weapon keen, etc.
The players should ideally get their items early, but not right away, as it gives them a sense of growth for both them and the item.
3) Try to make items more flavorful than just +1, +2, +3, etc. The minor properties chart is great place to look. You should also tailor each item for each character as well as each player. For example, don't give the newest player 15 different spells on his magic sword, it will just confuse him, and don't give the veteran power gamer an item that makes no sense for her character to wield just because "it's optimized".
They are a great way to encourage players to roleplay and seek out side quests and the like. Take advantage of it. Don't just make it Holy Avenger #532 and Robes of the Magi #42, you are building an item of legend! Give them as much backstory as your characters.
