A swordsman uses normal haki for their sword, something that is a must for a swordsman.
Headcanon.com, as stated where?
There it is, proof that ACoA destroys ppl from the inside out, now show me where it was stated that ACoA buffs the cutting ability of a swordsman.
Now I will go slowly because you do not understand my point.
Ryu is normal haki, something that was proven useless against Kaido:
And thus, you can not scar Kaido with something that was proven useless against him.
ACoA is internal destruction as proven in the panel above:
Which means that you can not scar Kaido with something that only damages the outside because a scar is something mostly superficial.
And yes, I agree with you, Zoro does not have ACoA because he lacks the ability to do internal damage.
So, no Ryu and no ACoA made Zoro scar Kaido.
Chapter 1010;
Title: Haoshoku
Main theme: Advanced Hao Haki.
Now we have Zoro, who used his last resort to attack Kaido, we know that
1. Normal Ryu is useless to deal serious damage to Kaido.
2. Zoro lacks the ability to use internal destruction (ACoA) plus the fact that you can't scar someone if you aren't even attacking his skin.
3. After getting scarred, Kaido mentions Zoro's CoC (note that there is no visible CoC burst so it wasn't basic CoC)
So, what's left, what's the last form of haki that we saw that it's capable of doing significant damage to Kaido? If you are clever enough you should alr know the answer.