D&D 5E Arcanist's Spellboook: What do you guys think of this homebrew feat?

I'm using the 5.0e standard for feat power ranges, rather than the more generous later 5e/One D&D power level.

That being said, how does the balance look on this feat? Is it something you would be likely to take or not take as a member of those classes? I like the old 3e guidelines for feat power levels: If everyone would take it, it's too strong; if no one would take it, it's too weak. Ideally, this feat (like all feats) should be highly appealing if it fits your character's non-mechanical concept (you should feel like you are being rewarded, rather than sacrificing power for concept), and not very appealing if it doesn't.

There are a few house rules to be aware of that this feat references. (Sorry for making it more complicated to judge the feat.)
-I allow prepared casters to prepare cantrips from their class spell list (or for wizards, from their spellbook) in their spell preparations. This is in addition to the ones that they always "know" from their class progression. I also assume that a wizard has all of his cantrips recorded in his spellbook. All wizards also start with prestidigitation in their spellbook, and therefore can prepare it if they didn't take it as one of their known cantrips.
-Sorcerers automatically gain one Origin Spell of each level from 1 to 5, based on their subclass. (This is for the earlier subclasses that don't already have a feature that grants them more than this.)

Arcanist’s Spellbook​

Prerequisite: Bard, Sorcerer, or Warlock
Through study and practice, you have unlocked some of the secrets of the wizard’s spellbook.
You have a spellbook, which you can use to prepare a single spell in place of one of the eligible spells you know from your class (chosen when you gain this feat). When you first gain this feat, your spellbook contains prestidigitation, three eligible 1st-level spells of your choice, and the known spell you replaced. Each spell that your spellbook contains must be both a spell of the class you are taking this feat for and a wizard spell.
You prepare your spell from your spellbook, and add spells you come across, exactly as a wizard does, except as noted. By spending the time and money you can also write any spell you know from your class into your spellbook, and choose to replace it when you gain a level using your normal rules for spell replacement.
Additional features are gained based on your class.
Bard. If you add one of the spells you learned from Magical Secrets to your spellbook and replace it with a different spell when you gain a level, the replacement spell must come from the standard bard list.
You can use an arcane focus as a spellcasting focus for your bard spells.
Sorcerer. If you add a spell you learned from your Origin Spells to your spellbook, you can choose it as the spell you replace when you gain a level and replace it with any spell from the standard sorcerer list.
Warlock. When you gain the Mystic Arcanum feature, you can choose to replace the spell gained with a spell preparation which can accommodate any spell up to the level of the Mystic Arcanum. You can add eligible spells of that level or lower to your spellbook. You can only have one of your Mystic Arcanum features replaced in this manner, and if you do so your original spell preparation can now only be used to prepare cantrips.

Note: The description of eligible spells is intentionally phrased to allow choosing spells that are class spells for you, even if they aren't on your normal class spell list.

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