Thats not completely true.
BCH has extremely low fees at the moment because noone is using it. BCH blocks are about 100 kB big currently..
In the early times of BTC you could send transactions with zero fee, which still got confirmed. Until blocks are full, no statement can be done.

'Faster payments' is a relative term. Block generation time is the same (~10 mins; BTC and BCH).
Payments are not faster in any of those two chains (considering same initial situation).

Segwit already increased the blocksize to 2MB (and up to 4MB) in a soft fork.
This also allows for more TX/s and lower fees.. But if people aren't using segwit, its their own fault.