Are you referring to timing attacks or something even more subtle?

Timing, data cache, BTB, you name it: doing software AES exposes you to All The Side Channels, and AES is somewhat notoriously hard to implement safely in software compared to other software-profile ciphers. This is the big selling point for ChaPoly.

Most/all software implementations of AES had various side channels in the past. Considering AES-GCM, as far as I'm aware no software implementation is considered "safe". Some libraries do not support AES-GCM without hardware instructions that make it safe (e.g. libsodium choose that way).

This is mainly due to AES relying heavily on substitution boxes, i.e. small arrays that are indexed with secrets, which is easy to implement safely in hardware, but difficult in software that runs on a processor with caches and such.

In the GCM case, it's also because the polynomial hash in GHASH wants fast polynomial multiplication, which PCLMULQDQ provides, and which you want (unsafe) lookup tables for otherwise.

Wouldn't a bit-sliced implementation of AES be guaranteed constant time, at the cost of some speed?

As long as you keep the keys not in HSM, that'll be true. Should also require the use of TPM for storing keys..

No, this isn't valid reasoning.