> TLS is supported in hardware, so it takes no more processing power than moving the data around without encryption.

Specifically, common PC CPUs support AES acceleration, so you can get the ongoing encryption for an established connection with minimal overhead. There is still some initial overhead for establishing the TLS connection due to the need to do key exchange (with asymmetric cryptography primitives typically based on Diffie-Hellman with finite fields or elliptic curves).