You are the DM. You both know the rules and are the arbiter of what the characters/players do. You decide what works and what doesn't.
In terms of the player, there are NO sources that your player can cite in 5e that allows them to take the actions described. They do NOT exist in published official 5e content that would be available to a level 7 character. Allowing the player to behave as if they can do these things is simply allowing them to cheat or at the very least just use house rules they may have dredged off the internet. You are the DM. The rules available in a game you are running is up to you and not the players.
You mention "This is not about sitting down with him and having a serious talk.....that's been done. " I don't understand that statement. You told the player that they are cheating and that the rules they are relying on are NOT part of 5e and that you would allow them to have the zombies or skeletons that the spell allows them to create and not the other forms of undead that they seem to have produced by citing rules that do not exist in this version of the game ... and they said No?
You can then tell them that they aren't welcome if they are going to continue cheating. Trying to solve this problem in game by vaporizing the undead via some in game occurrence isn't going to work. Once the undead are gone, and you prevent any more from being created, the player may simply decide that you are picking on them (since you are) ... and then will quit anyway ... so you might as well get the issue resolved up front, with the entire group, not just the one player, and figure out a path forward that works for everyone. A path that doesn't involve all of the higher level undead and ignoring the rules of the game you have set up.
It is important to note that there is absolutely nothing wrong with running a game where necromantic magic is enhanced so that it works differently. Perhaps it can even work the way the player wants it to. However, that isn't the player's descision, it is yours as DM. You decide the rules under which the game world functions. Ideally you decide those rules at the beginning (and it sounds like you decided on the 5e ruleset without homebrew extensions). After that, it is up to the DM to decide if rules change - not the players. It is pretty much part of the fundamental agreement that players and DMs enter into when they start playing together. The character/players decide their actions within the game world while the DM adjudicates those interactions using the logical structure provided by the rules.
Anyway, I suggest having a talk with the entire table, explain how these necromantic spells work in 5e and what sorts of dead creatures can be targeted and what sorts of undead are produced and then discuss how best to address the issue for your table and group of players. Explain that the necromancer player has been mistaken about how they work, that you weren't sure yourself and went along with it for a while but that you have now looked it up and the spells do not work as the necromancer player had thought so you are going to retcon the situation to fit within the way the rules should work. You can either allow the necromancer to keep the undead they have but any further undead creation WILL follow the rules, or you could change the undead the necromancer has to either skeletons or zombies - give the player the choice as to which they want to have, or you could just take them all away (I'd avoid this option because the mistake in allowing it to get this far is primarily the DMs, the player should have never been able to create these undead in the first place but you can't blame them because you said yes or didn't stop them when they first created the undead).
If the player can't accept the adjustments then they have the choice to leave, they can be asked to leave if they don't accept gracefully, and you can choose to leave if the player makes the game feel toxic after you fix the situation.
You are the DM. You both know the rules and are the arbiter of what the characters/players do. You decide what works and what doesn't.
In terms of the player, there are NO sources that your player can cite in 5e that allows them to take the actions described. They do NOT exist in published official 5e content that would be available to a level 7 character. Allowing the player to behave as if they can do these things is simply allowing them to cheat or at the very least just use house rules they may have dredged off the internet. You are the DM. The rules available in a game you are running is up to you and not the players.
You mention "This is not about sitting down with him and having a serious talk.....that's been done. " I don't understand that statement. You told the player that they are cheating and that the rules they are relying on are NOT part of 5e and that you would allow them to have the zombies or skeletons that the spell allows them to create and not the other forms of undead that they seem to have produced by citing rules that do not exist in this version of the game ... and they said No?
You can then tell them that they aren't welcome if they are going to continue cheating. Trying to solve this problem in game by vaporizing the undead via some in game occurrence isn't going to work. Once the undead are gone, and you prevent any more from being created, the player may simply decide that you are picking on them (since you are) ... and then will quit anyway ... so you might as well get the issue resolved up front, with the entire group, not just the one player, and figure out a path forward that works for everyone. A path that doesn't involve all of the higher level undead and ignoring the rules of the game you have set up.
It is important to note that there is absolutely nothing wrong with running a game where necromantic magic is enhanced so that it works differently. Perhaps it can even work the way the player wants it to. However, that isn't the player's descision, it is yours as DM. You decide the rules under which the game world functions. Ideally you decide those rules at the beginning (and it sounds like you decided on the 5e ruleset without homebrew extensions). After that, it is up to the DM to decide if rules change - not the players. It is pretty much part of the fundamental agreement that players and DMs enter into when they start playing together. The character/players decide their actions within the game world while the DM adjudicates those interactions using the logical structure provided by the rules.
Anyway, I suggest having a talk with the entire table, explain how these necromantic spells work in 5e and what sorts of dead creatures can be targeted and what sorts of undead are produced and then discuss how best to address the issue for your table and group of players. Explain that the necromancer player has been mistaken about how they work, that you weren't sure yourself and went along with it for a while but that you have now looked it up and the spells do not work as the necromancer player had thought so you are going to retcon the situation to fit within the way the rules should work. You can either allow the necromancer to keep the undead they have but any further undead creation WILL follow the rules, or you could change the undead the necromancer has to either skeletons or zombies - give the player the choice as to which they want to have, or you could just take them all away (I'd avoid this option because the mistake in allowing it to get this far is primarily the DMs, the player should have never been able to create these undead in the first place but you can't blame them because you said yes or didn't stop them when they first created the undead).
If the player can't accept the adjustments then they have the choice to leave, they can be asked to leave if they don't accept gracefully, and you can choose to leave if the player makes the game feel toxic after you fix the situation.