If said schools are not available to PCs my first question is "Why the hell not?" and if the answer is for power reasons my second question would be "If those spells are that unbalancing why are they even a part of the setting?".
Because they are cool, and show the power of antagonists. Avada Kadavra ! is totally unbalanced and very, very scary, obviously a NPC only spell. Still it exists and it's at the core of what makes the story as good as it is.
By the second question, I mean that if a school of wizards out there have access to these too-unbalanced spells why haven't they made better use of them by taking over the setting or whatever?
But they are, actually, they are the threat ! Of course, to be stopped by a band of unlikely heroes, come on, it's the core of the genre !
Has the DM taken a broader view and asked what these spells in the hands of unscrupulous NPCs would have lead to in the setting over the long term?
Obviously, see the example above.
This is why a DM can't just throw this stuff in "for fun" without giving it some real thought. Everything has broader implications and affects on the setting.
But it does. Look at Lord Soth or Voldemort or Stradh, they have had setting-wide implications.
That said, if you don't care about the setting's ongoing believability then all this is fine.
I'm sorry, but this is a very low blow, and totally unjustified. The fact that a few NPC are very powerful individually is certainly taken into account in the setting, and in the story, as is the manner in which it influences the story.
You have absolutely zero reason to denigrate other points of view like this, both the fantasy genre but also most of the settings show that you are wrong in this.
If they're too powerful for PCs to have they're too powerful for NPCs to have. Remember, the NPCs who have this power can and likely will also have been using it "again and again" all the way until they meet the PCs - and even beyond that, if the PCs lose or the NPCs flee - with whatever downstream effects that may have on the setting.
First, it's absolutely untrue, a Death Knight having Parry at will will not make a huge difference on the setting. The Hellfire Orb does not make him more dangerous than a lot of lower level wizards. It does, however, make him scary for adventurers on one encounter, and his evil influence is well felt on the setting as well.
Pun-pun arose because 3e gave too many powers to everyone and as a result a few broken combos slipped through the cracks.
That is true, but the real reason of pun-pun is using the sarrukh power, which is not and should never have been available to PCs, whereas on the other hand, it makes perfect sense in the setting for the domination of scaly creatures in their era. And YES, it was totally integrated in the setting with its consequences. Is that a reason for giving it to PCs, obviously not.
Have less powers for everyone - PC and NPC alike - and the chances of a pun-pun error drop dramatically: it's easier to check for broken combos.
Unfortunately, it's the "play only at low level" answer, which is not satisfying in a epic game like D&D. But if you want to make full use of all the levels, you can certainly do so, but it requires some control.
Also, pun-pun can't get into the hands of a player if Kobolds and other monsters can't be PCs. Errors upon errors, this took.
And yet we have kobolds PCs officially in 5e, but no punpun because some powers are for NPCs/Monsters only.
Your players are putting reasonableness ahead of advocating for their characters.
And they have no right to "advocate for their character", they are there to play the game, amongst friends, with all the same intent, to have fun. They are not there to nitpick and whine and confront the DM with demands.
A character advocate's first response would often be "What would it take for me to be able to do that?", simply because what's good for the goose is also good for the gander.
And the answer would be: Why don't you go ask Zariel to see if she can do that for you, but if she does, you will become an NPC, since you will hae decided to completely depart from the flow of the campaign. Where is the problem in that ?
And there's a flip side as well: there's no such thing as PC-only powers and abilities. If a PC can do it, an equal NPC can do it also.
Not necessarily, as a DM, why would I ever claim that right ? I can do whatever I want, and it might be cooler and more appropriate to giving my players a good time.
I get this, but at the same time if I think an opponent can't be made worthy with the abilities available to it (and to PCs) the answer isn't to just give it more abilities willy-nilly, the answer is to add more opponents!
Of course, rather than one Voldemort with Avada Kadavra, let's just put 20 incompetent death eaters instead, it obviously does the same effect.
The DM is - or should be! - constrained by the setting every bit as much as the players; and here "the setting" includes spells and abilities extant within that setting that are not specifically innate to a species.
That is your opinions, but if I may, it's a purely gamist view, which is your own and certainly not supported by either the genre (see all my examples above) or by the rules.
Listen, I'm sure you are a great DM, but you are working using constraints that work for you and your table and that are not necessary for many other gamers (or actually writers of the genre) to create extremely entertaining media for other people. It does not give you any right to impose these constraints on others and to criticise them for being inconsistent, we are also very consistent but using different paradigms than yours. And although you could certainly say that creativity flourishes under constraint, there will be at least the same amount of voices telling you that not having constraints and having more creative freedom is great for them. After that, it's just a matter of taste, and to each his own.