Still don't care about the spell school. I have no problem with whatever one you want to stick it in.
Doesn't even need to be a "spell". Elemental monks can deal a variety of damage, while pushing or pulling.
When you hit with an attack, spend a swordmage point to do a swordmage smite... or whatever.
On a hit deal cold damage, and its speed is reduced by 10 feet until the start of your next turn
Litterally Ray of Frosts effect, just with a weapon.
Smite of Frost.
Black Pudding disagrees
https://www.dndbeyond.com/monsters/16808-black-pudding
This is fantasy magic. It can have fantasy acid.
I haven't suggested anything a wizard isn't able to do. (Well, the acid is new, but still on theme).
Ray of frost, with a sword.
Thunderwave, with a sword.
Swordmage is a mage, with a sword.
You mean like a paladin?
Or an elemental monk?
Not sure why you think they would get "every single" one of them. Both 5e wizards and 4e Swordmages only got to choose a few options out of the list.
Similar, sure. But not a strict copy.
It still needs to be within the 5e framework, and balanced with the 5e classes. Subclass at 3, ASI at 4, big damage boost at 5, ect.