I have half a mind about this. I do think Social Links impacting the party member's combat prowess is a good idea. On the other hand, I don't like the idea of levels being removed or combat progression being solely tied to Social Links because there's just so much about it that could go wrong. Like, you might not have much time to improve some of the links due to scheduling concerns which can nerf them. If reversals and broken links are used more (and I recall numerous people supporting this), you can potentially screw over a party member by mistake. I dunno, there's other stuff too that I can't think of up the top of my head. Frankly, I think P4G got it right when it came to Social Links strengthening characters, even if the balance was off. Have them learn new skills the higher the link is or something or give them a stat boost. I don't think you need to ditch character levels.
Honestly most of your ideas would work better like this: Let's say the protagonist's initial persona is special and cannot be dismissed or used in fusion; in other words, it's a permanent part of the persona stock. This persona does learn some moves on level up but only a limited amount and up to a certain point. The persona can learn new techniques or get stat boosts or elemental changes based on how far in social links. Like, maxing the Magician link teaches Mind Charge for example. The persona evolves when you max all the links. Now let's also say you can determine your (and by extension your initial persona's) arcana at the beginning of the game in a similar way to Tactics Ogre. If a Social Link that matches your arcana increases, you get an additional stat boost or some other benefit that you wouldn't have gotten if your arcanas didn't match.
Something like this would work.