The client that sends SNI is, AFAIK, the browser or a similar piece of software. Some older browsers don't support SNI so they can only access single-vhost-per-ip over https.

This means you'll have a really hard time trying to get rid of SNI system-wide, what with a lot of minor apps making their own https connections (granted, on Android or iOS they probably use a common API, but not on a computer).