When you want to remove and reuse a waystone from your list, but only have a silk touch pickaxe on you, you actually can't.
Because the silk touch will always return the named waystone, even if you've deleted the name from the list prior to mining the waystone.

So in the case you want to delete a waystone, you have to use a fortune/unenchanted pickaxe to be able to remove the assigned name. This is strange, because why is there a delete button in the list of waystone locations if it doesn't help here?

The delete button only removes the waystone from your list, other players may still have it in their list though (assuming multiplayer). So it's not meant to be an "invalidate waystone"-kind of delete.

Could possibly make silk touch only actually silk-touch if the player has the waystone activated, but I'm not sure if that's really the way to go, since that adds a hidden complexity to the silk touch enchantment that doesn't really match the expectations players may have from Vanilla's silk touch behaviour.

All right, I see. Understood, indeed don't want that. This is for single player though.

But what if it does what I suggested but only after the waystone has been "removed from list" for all players? As in, doesn't exist for anyone anymore, so the name doesn't either and waystone drops as unnamed?

This would effectively mean in single player that every delete indeed actually removed the name from the block as there is only the one player holding the name in their list when they remove it.
I'm not sure what happens if anyone is holding a named waystone in their inventory though

I would rather not add any special casing here, as it would go against the nature of Silk Touch. You would run into similar problems trying to get cobblestone instead of stone while only carrying a silk touch pickaxe, so I believe this is an issue that should be solved in normal gameplay.

I have an alternate suggestion that would still solve this issue, so if you agree (and if it's possible), please reopen.

When you choose the Delete option in the menu, it should still remove the location from your personal list as normal, but it should also attempt to permanently delete the location globally if you were the last owner. Then effectively the delete button already normalizes the waystone (revert it back to unactivated) if only one person had it on their list (you), so you can safely use silk touch without maintaining the data you wanted to delete (since it's already gone by then).