Yes. If you magic initiate into your own (“spells known”) class, you can pick up an additional known spell of first level, thus saving a slot. Other than this exception, I would tend to keep class feature spellcasting and additional feature (race/feat) spellcasting separate.
Can you take the magic initiate feat more than once?
You can take each feat only once, unless the feat’s description says otherwise. You must meet any prerequisite specified in a feat to take that feat. If you ever lose a feat’s prerequisite, you can’t use that feat until you regain the prerequisite.
Does Magic initiate scale?
The Magic Initiate feat, as you’ve correctly noted, lets you learn two Cantrips, and Cantrips scale with your character level, not your caster level. This means that your Initiate-Rogue can get the full, scaling power of their Cantrips for the rest of their life!
Does Magic initiate level up?
Can a magic initiate be used in any class?
Again, even for spells known classes, the spell magic initiate gives is associated with a class (the one you choose), and if that doesn’t match your spellcasting class, that spell wouldn’t count as a spell you can use slots on (per the precise wording of the spellcasting feature of your class).
Can you cast a 1st level spell with the Magic initiate feat?
Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell. If you have spell slots, can you use them to cast the 1st level spell you learn with the Magic Initiate feat?
Can a warlock use the feat magic initiate?
If a Warlock takes the feat Magic Initiate (Warlock), it knows one more 1st level spell from the Warlock spell list. Still, the Warlock has only 1 spell slot (at 1st level) from the Warlock table, but it can use the feat Magic Initiate you cast only that one spell as 1st level spell, once per long rest.
Why are spell slots used for magic initiates?
Your interpretation overall is flawed because one of the things that keys off of being “prepared” by a class is the ability to use spell slots to cast a spell for such classes.