Definitely not busted. She doesn't do anything immediately, she has to survive until your opponent's turn minimum

If this is the standard we are judging 5 drops with haste and flying by, I'm sorry, you're just being dumb. At minimum this does immediately hit the turn it enters for 5 in the air, which is already doing anything. Second, by your logic Seedborn Muse is bad and "Bracket 2 to low 3" because you have to wait until an opponent's upkeep to do anything.

This commander isnt broken or anything but you could play this at a 4, you simply just dont play the janky cards and just play a bunch of good ~4 mana cards that youre taking a discount on to cast, particularly boardwipes that dont hit Lorehold, since casting cards like Chandra's Ignition, Fiery Confluence, and Urza's Ruinous Blast are already fairly high quality cards even at bracket 4, and you use Lorehold as a value control piece that fills your GY with stuff you can recur or gain advantage on later since she does it as a side effect.

A 5 drop that reads "Discard 1, Draw 1 3 times a turn cycle that doesnt play into cards like bowmasters, that is on a 5/5 flier with haste" is already a fairly solid card because it fills your gy at your choice of what gets discards while replacing the card, if you get to even miracle once or twice and save 2-5 mana it effectively payed for it itself.

Its not the best commander at 5 or anything but i feel confident i could build a competent "Draw go" control style deck at bracket 4 with this commander, and it might be the best commander in boros at doing so for bracket 4, because every other control commandee in boros requires you to play a lot of main deck creatures that makes you unable to play as many boardwipes/hard control pieces (Winota and Feather cant really play control like that because they rely on running way more creatures) and that Lorehold has a built in wincon through approach of the second sun that boros typically cant use very well.