It's a presentation thing. If you can get class features with a feat, then it's kind of a light sprinkle of multi classing, which is fine. But when your class feature is 'you get a feat', even if from a limited list, then it's no longer your class feature, it's just a feat.
Magic Initiate isn't the Spellcasting class feature.
Is there any other class in the game where there is a feat letting your steal from a Subclass the same way you can pick up Battlemaster maneuvers? I can't think of one...
You can get the maenuvers with a fighting style too, so it is not just a battlemaster thing, but there plenty of examples from other classes.
Mobile feat steals from Swashbuckler subclass.
Poisioner steals from the Assassin subclass
Magic Initiate allows you to pick up spells and cantrips otherwise only available Wizards in the Dunamancy or Chronogy subclasesses.
Ritual Caster is a ripoff from pact of the Tome Warlocks
That is just subclasses, you also have the following class ripoffs:
Gift of Chromatic Dragon is close to being a ripoff of Elemental Affinity from the Draconic Sorcerer subclass.
The fighter unarmed fighting style, the fighting initiate feat that lets anyone get that, Tavern Brawler feat, Dragon Hide feat and several racial abilities all steal the unarmed strike ability from the Monk.
The skill expert, prodigy feat steal expertise from Bard and Rogue and in addition to those two, several backgrounds and the skilled feat let you take thieves tools from the Rogue.
I don't get your comment about Magic initiate, because it does allow any class to cast. Telepathic feat, Fey Touched Feat and Shadow Touched feat all do that as well, stealing from multiple casters.
The metamagic adept feat steals from sorcerer
The invocation feat (can't remember what it is called) and Warcaster both steal from Warlock