On this point, the goal is exactly, "A PC can pick a Level X monster and player it in a part with Level X players."
The reason is, Level (of monster) must be the same as Level (of player character). There needs to be a single of unit of measurement that the 5e game engine uses consistently and meaningfully. This useful use of "Level" as the measure of power of any creature helps the DM understand clearly and precisely how much power they are introducing when building a combat encounter against the player characters.
Level = Level = Level
No, that is a bad idea.
A level X PC is designed for a single human being to spend a large percentage of their attention managing and controlling.
A level X monster should not be designed for this.
Attempts to make PCs simple enough that a DM would want to control 5 of them make PCs too simple; players want more levers and bells and whistles.
Similarly, DMs shouldn't be pulling the levers and bells and whistles of 5 complex tokens; it is a waste of the DM's mental effort, and it doesn't make encounters more fun. DMs have better things to do.
PCs in 5e are not designed to fight against other PCs - the defence vs offence balance isn't tuned. Pacing is just whackey. I mean, you can try to do it, but it makes things worse. And quite possibly making it happen would make other desirable properties of the game not work, like how PCs interact with higher and lower level foes and the like.
If you succeeded at the described task - made a monster template that could be swapped for a PC template - the result
demonstratively sucks, as
within lived and tested experience we can see that fights against 5e PC-build monsters aren't really that fun.
So you are
first talking about having to rework the entire 5e PC side of things,
then you have to solve for pacing issues, work out the power curves that give the experiences you desire with relative foe levels, etc.
...
I'll give a concrete example.
If PC damage starts out at 10 DPR and goes up by 3 per level, while PC HP starts out at 10 and goes up 6 per level.
PC accuracy goes up by 1 every 2 levels, while PC AC goes up 1 every 4 levels.
If we want fights to last ~3 rounds, drain PCs of 50% of their HP on average, and PC accuracy against a level 1 foe is 50%, and level 1 monster accuracy against level 1 PCs is 30%, then level 1 foes should have ~15 HP and do 5 damage on a hit.
At level 10, we might expect 60% accuracy and monsters hit 50% of the time; PC have gained +4 to hit and +2 AC, while monsters have gained +6 to hit and +2 AC. PCs have 37 DPR before accuracy, and have 64 HP. The same 3 round pacing puts even level monsters at 21 DPR and 67 HP.
At level 20 we might expect 70% accuracy and monsters hit 60% of the time. PCs have gained another +5 to hit and +3 AC, while monsters have gained +5 more to hit and +3 more AC. Players have 67 DPR before accuracy and 124 HP. The same 3 round pacing puts level 20 monsters at 141 HP and 35 DPR.
L1 PC: 16 AC, +5 to hit, 10 HP, 10 DPR
L10 PC: 18 AC, +9 to hit, 64 HP, 37 DPR
L20 PC: 21 AC, +14 to hit, 124 HP, 67 DPR
L1 monster: 16 AC, +1 to hit, 15 HP, 5 DPR
L10 monster: 18 AC, +7 to hit, 67 HP, 21 DPR
L20 monster: 21 AC, +12 to hit, 141 HP, 35 DPR
This model makes monsters that are
designed to fight level X PCs, not
monsters designed to emulate level X PCs. We can compare threat volumes of monsters of various levels and check if we get good experiences when (say) a level 20 PC fights a bunch of level 1-3 monsters.
If we are taking PC development as a given (say, we are making our own monsters for 5e, but leaving PCs alone), then we calibrate the PC model and derive a monster model from it. We have a number of degrees of freedom in the monster model - we can tweak how much of its offensive power comes from +ATK and how much from damage, and similar for AC/Saves vs HP - as those have a large impact on different level PCs interact with a given monster.
If we have the freedom to change the PC model, we get more degrees of freedom.
For example, we might want monsters to gain HP faster than they do damage output, and PCs go gain damage output faster than they do HP, in order to (a) make low level monsters feel threatening but weak, easy to clear out but dangerous if ignored and (b) make high level monsters feel oppressive but not instant-death for lower level PCs. We can do this while maintaining the same combat pace against even level monsters.
That might create tension you might like: if 50 low level soldiers are aiming at you in a PC-defence-scales slowly model, you are threatened even at higher level. Meanwhile, if you face up against a dragon too early, you won't be able to take it out because monster-defence-scales-fast. If you are anywhere approaching the right level you won't instant-die from the first attack, however. (It will hurt a pile!)
In any case, my point is that making monsters
designed to fight against PCs in fun ways is a different problem and opposed to
making monsters that scale just like PCs.