Is ChaCha20 safer than AES with respect to side-channel attacks?
It basically depends on what you consider side-channel attacks.

If you consider time/cache side channel attacks than chacha20 has been design with resistance to such attacks in mind while AES didn't. 
In fact, AES is vulnerable to these kind of attacks (as they were invented after AES was designed). But, hardware implementations, such as AES-NI are protected because they don't use caches and execute in constant time. 

If you consider also power/electromagnetic side-channels then both algorithms are vulnerable and in this sense the community has researched A LOT more on protections for AES then for Chacha20 which will be significantly harder to protected due to its ARX nature which require expensive masking conversions. 
So it will be more easy to find protected , pure sw, AES implementations, even in open source. But you need a protected implementation, otherwise you will be vulnerable.

To summarize you have to check which side-channel attacks you want to protect against and the security claims of your implementation. It can't be simplified as "AES needs to use AES-NI" because we can have side-channel protected AES in pure software (see the link above).