Yeah, there's a rock paper scissors element to this house rule. If you have the right spell or feature for the right enemy, then the effect goes through easily--but if not, too bad. It has a kinda witcher-y element to it, where identifying enemy weaknesses is important. That probably necessitates thinking about the way encounters happen, so that PCs have the opportunity to figure out what they should be doing. It also creates meta-game knowledge issues, advantaging players who know the MM.
For those PCs that don't have a lot of flexibility in what they target--it might be fine. If the balance of easy to hit targets and hard to hit targets is reasonable, it might just mean that gameplay is very feast or famine. I don't have a sense of how much fun it would actually be to play with this kind of setup--it might make being a monk just... miserable. Features that target constitution and strength suffer a lot more than others--since monsters usually have good scores--maybe the simple solution would be to have monks' stunning strike and other similarly restricted classes' features target something else.
It's kind of like rock paper scissors, but only if many of the participants are restricted to only throwing one or two of the three, while the DM can throw anything they please (having the entirety of the MM to draw upon).
IMO, it's not usually that hard to figure out a rough estimation of a creature's stats. Big, bulky creature - high str and probably con. Quick on its feet - dex. These are readily observable. The mental stats can usually be inferred based on a quick conversation with the creature, assuming the DM RPs it remotely accurately. If it can't talk, odds are pretty good that it's at the lower end in terms of mental stats. Animals pretty much invariably have an intelligence in the low single digits.
Assuming that the DM isn't the type to obfuscate what ought to be obvious and observable, no meta gaming is required, IMO. Obviously, this does assume a reasonably experienced group of players; newbies might find it more challenging to translate descriptions into mechanics.
I think the biggest challenge is to determine a fair way to design encounters within this new paradigm.
For example, let's assume you have a caster who likes to cast Hypnotic Pattern. Based on a quick flip through of the MM, it seems like most creatures have between 6 and 13 Wisdom. That's child's play for a mage with an 18 casting stat and +3 Prof mod (+7 bonus) under your proposed system, as I understand it (a single attack roll at +7 against what is probably around a Wisdom of 10).
Under the RAW system, those 10 Wisdom creatures might not have a great chance to save vs a DC 15, but it's significantly better than your system (30% vs 10%), plus each creature gets an individual save meaning that odds are decent that at least some in the group will make it.
So do you pick creatures at random? If so, then the typical Wisdom will be 10 and Hypnotic Pattern will be VERY strong (significantly better than RAW).
Do you pick only creatures with a high Wisdom to challenge them? The players might perceive that as unfair (and there's an argument that they're right).
It's not as bad if you redesign the classes to have the flexibility to target multiple different saves, but even then you'd probably also want to redesign monsters to have a narrower range of stats.