A slot system is a fascinating idea, and I really think pursuing that line of thought might lead to some interesting ideas.
Int score slots I don't think is good, but I like using Int Mod + PB as a base. And, frankly, I think there needs to be some consideration for included features that don't take up slots. For example, there's no reason that a caster's Spellcasting feature should take up a slot, especially if cantrips and spells will. I also don't think its worth it for racial traits to take up a slot because most racial traits aren't worth anything. If a Lizardfolk's natural weapons has to take up a slot...well, that's not a game I'd personally be interested in playing.
You'd have to look at MOBAs for your model, not MMOs. MOBAs have usually 4 slots for powers + a passive. Sometimes, the 4 slots for powers also are loaded with various passives too, giving more than 4 abilities. I'd say it'd make the most sense for a character to start off with either PB x2 or PB + Int mod at start, and then at 5th, 11th, and 16th level, they get a bonus number of slots.
In order for this kind of system to really shine, you'd probably have to redesign classes. As they are right now, I don't think slot-based design would be that interesting. I could be a mid-level Ranger receiving Wisdom saving throws while the party Wizard is choosing 4th level spell slots. We can argue all day about whether this is equal or not, but it doesn't feel good -- saving throw proficiencies, while useful, don't make fun at the table, just let you have more fun by getting you out of trouble. So, I'd say that any and all proficiencies, including Expertise, should not take up a slot.
Likewise, at higher class levels, this option does nothing for Fighters, barbarians, or monks as they are now. Brutal Critical taking up a slot would feel like a slap in the face, as would Brutal Critical's improvements. Likewise, Timeless Body for a Monk taking up a slot feels likewise pointless. After all, I'm getting Timeless Body and the Cleric gets Earthquake? Mid.
If you were to redesign classes, it'd be wise to look at how MOBAs handle simple vs complex champions and use that to mirror low-level features and high-level features. At low-level, you get features that do usually one thing, and can be used in various ways but remain simple. This would be something like Action Surge, or Pact of the Chain, or Burning Hands. At high level, you'd want a feature to both have an active and a passive component. Maybe a Barbarian gets a Strenght-based feature that when activated lets them throw Large & Huge objects/creatures, and passively lets them grapple these creatures or lift x20 their Strength score at a time.
As for how many slots you end with, we have to think exactly when is it a lot of players have trouble keeping up with their features. Now, anyone whose run a game for new players knows that some people will have trouble with their features forever, and others will take to it like water. But I'd say, once you have 5 different levels of spell slots, and once you have like 15+ spells + class features, you're probably nearing your "mental limit" for how much you're likely to remember at any one given time. With this in mind, I'd say probably around 8 active slots would be good, and then you have 4 things you can maybe use a Bonus Action to swap into your active rotation. This gives you 12 effective abilities and you can tactically play with them, mixing and matching different combinations to achieve different effects.
However, the more I think about it, the more I wonder if it might be better to try this with just a completely different style of game. Maybe having just 5 slots and being able to change them in and out would feel better, and then you'd have a lower level limit, maybe between 1-10 for example.