Is it illegal in the US to offer remittances using cryptocurrency, and can crypto remittance services avoid the fees that regulated remittance providers pay?

There is nothing illegal about remittances being performed, using crypto-currency as the value of choice. If both parties are willing to accept crypto as form of remittance, that is perfectly okay.

What is however required is that you have a money transmitter license in all the US States in which you will be signing up a customer from and that you conform to the federal BSA laws, CFPB laws and others mandated laws like the Patriot Act, etc.

You can use smoke signals, crypto-token, pigeons, etc. all as long as all the parties in the transfer chain (flow of funds) are licensed and agree to it.

On the second question, ofcourse they can side-step, but then you would have your own costs (usually compliance, etc.) that your users would have to pay.