> Every memory begins with tiny changes inside the brain

Maybe. But the brain is not the only place where memory is stored. Flat worms remember things (and skills!) after their head has been cut off and they regrew it:

https://www.smithsonianmag.com/science-nature/these-decapita...

Flatworms branched off our side of the animal tree of life very early on. They're on the same side as molluscs, some of whom (cephalopods) are famous for having a more distributed nervous system.

Granted though many/most organs are stateful and somewhat adaptive - in a sense they'll "remember" what happened. Even plants possess that to varying degrees.

At least spinal cord has a kind of memory related to movement, but that's something else than episodic memory obviously.

I think the evidence is strong, here. Quite difficult to form new memories without a brain!