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Actual implementations of 256 bits AES are already breakable using side-channel attacks.

so if that key size is increased, would these side channel attacks be either mitigated, or made impractical?

No that's not the way to go for countermeasures against side-channel attacks (breaking 4096 bits key in asymmetric crypto is also doable using side-channel attacks, even in the cloud [1]). Since it is a physical attack rather than a mathematical one, the countermeasures are at the physical level too: the idea is to make the cryptosystem leak less information through channel such as power consumption, computation or memory access time, electromagnetic radiation… Currently most of these countermeasures are at hardware level.

[1] https://news.ycombinator.com/item?id=4865516

Edit: forgot to add reference link.

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