The idea is that any "legendary" (or epic, or solo, or whatever the label is) creature needs "three strikes" from a given spellcaster (that resets if the monster lives to take a short rest). Each time the monster fails a save against a sufficiently high-level spell, that counts as one strike. (For a low-level legendary, this might be as low as 3rd level. For monsters of the highest CR, it would be spells of 6th level or higher, the ones where you only get one per level, to reinforce the "specialness" of those).
Only spells with "strike" info counts. For instance, Fireball would not have strike info. You would not be able to cast two Fireballs against a Legendary Wyvern and then top it off with a single save or suck spell that does have strike info. This is because spells that primarily deal damage has that utility anyway.
For instance, let's invent some strike info for Forcecage:
Forcecage
7th level Bard, Warlock, Wizard Evocation strike spell
[regular info here]
1st strike: The monster visibly struggles against the invisible prison, rendering it effectively grappled.
2nd strike: The monster is losing the fight, barely being able to keep gaps in the prison open. It is restrained.
3rd strike: The spell works as described. (This part is always the same)
Strike info is ignored for regular versions of the monster.
This way, a single save-or-suck spell doesn't end a Solo encounter. Yet, it is hardly pointless to cast these spells anyway, since you do get benefits from every (successful) casting.
Since hit point totals are increased for Solo monsters (rather massively, I might add), the party can't just drop the idea to cast debilititating spells and "just kill it". The idea is that the monster should last long enough for a challenging combat, no matter the party's tactics (causing damage, causing it to "suck", or both).
And yes, the idea is for such an encounter to actually challenging on its own, right off the bat, no DM tweaking needed. No expectation is made on having any other encounters that adventuring day!!! (If there are more encounters, that makes the Solo "too difficult" rather than being a prerequisite for being dangerous at all).
Hmmm, instead of "Legendary Save" you fail, reduced effects, ... I kinda like it.