I don't think this part is true. There are constant time software implementations of AES: https://crypto.stackexchange.com/a/92/21442
(Designing easier-to-implement-securely algorithms does help.)
This attack is reading data directly from the bus between RAM and the CPU. You can not make an algorithm that survives that.
ChaCha20 is sufficiently constant-everything (which includes not having any key-dependent RAM access patterns) that we'd probably need to pick up the data (not just address) lines. That turns out to be (mildly?) harder in this particular combination of attack target and measurement setup.
We do make appliances designed to survive (or at least strongly resist) such attacks, but admittedly we don't rely on naive software AES implementations operating on external RAM. ;-)
As technion says, ChaCha20 was designed such that the evident software implementation resists such attacks; however, Schwabe and Kasper also have a high-quality software implementation of AES.
Hardware implementations are a different beast altogether, and a lot of expertise has gone into making hardened AES implementations in hardware (as forg0t_username says, masking helps - but this is an entire field of study. Look at some CHES conference papers to get an idea.)