With the name based shared hosting, there is one problem, they will not work with browsers that do not send the hostname as part of requests.

Well, this is true only for older HTTP/1.0 browsers that have not retrofitted the host field feature from the HTTP/1.1 protocol. (The “Host” header that distinguishes between various DNS names sharing a single IP address was optional in HTTP/1.0; it is mandatory in HTTP/1.1, issued in 1999 as RFC 2616)