Yes, a wizard would take this feat. In fact, a wizard should take this if it was two feats; they should just pick variant human and take it at 4th level.
It's that good.
If you're going to homebrew, start by deciding .... what problem are you solving?
And you haven't done that. As far as I can tell, this is just ... cool to you. And there's nothing wrong with cool! Cool is awesome! But there is no reason to include it ... it doesn't fix anything.
So, after understanding the why, you then have to think about the bigger issues:
1. Does it work in a way that is congruent with 5e's rules, such as the action economy?
Your spell doesn't fit within 5e's rules. It doesn't have a trigger, like counterspell. There is no way to just have a "reaction tax" on a spell. In addition, it is very much blinkered to have an action / reaction / (maybe action). And you can't just say free action (as you did before). Best to use the standard "action" and work from there.
2. Is it similar to some other power/ability/spell, so I can judge how powerful it is?
Let me give you an easy example of this. Let's say I think that Bastard Swords are cool. So I created a new Bastard Sword that had the following properties:
Damage 2-16, finesse, one-handed
Now, given that the closest comparator is the rapier, everyone would immediately say that this homebrew is overpowered; in effect, I am increasing the max damage for a finesse weapon from d8 to 2d8.
Now, your closest comparator in the 5e PHB is Wish. Wish is a 9th level spell. According to the PHB, "
Wish is the mightiest spell a mortal creature can cast." And what does it do? What your spell does ...
"The basic use of this spell is to duplicate any other spell of 8th level or lower."
That's right; you have created a second level wish spell.
For a Wizard, you have moved Wish from 17th level access to 3rd level access.
3. Understand how homebrew fits into the design ethos of your game.
The two issues which people have noted are:
a. As it is, spellcasters are generally privileged over martial characters in 5e. As such, any ability that gives them even more power should be viewed as possibly unbalancing. Especially when you are dealing with a spell that is, effectively a form of metamagic.
b. The need to pick and choose spells (resource management) is one of the few constraints on spellcasters in 5e. Obliterating that distinction needs to be done very carefully.
Finally, after these considerations, it helps to see what other people have done. For example, a quick search on this forum which I just did turned up the following:
Okay, so bear with me here. I'm working on a 5e-appropriate conversion of Rary's mnemonic enhancer that works within the 5e magic system. Here's my first draft: Rary's Mnemonic Enhancer 4th-level transmutation (wizard) Casting Time: 10 minutes Range: Self Components: V, S, M (a piece of...
www.enworld.org
That was a homebrew attempt at Rary's Mnemonic Enhancer; compare it to your homebew.
I don't think you are going to get much positive feedback, because you are far too invested in your idea to see the evident issues. So I really suggest you just play it. If it works for you, great.