Being connected to the internet means being able to talk to every single machine on that network. If a computer can't talk to IPv4 computers, then it is by definition not connected to the internet.
We can't split the network. If we have to do it to switch to IPv6, then it just won't happen. Ever. To avoid splitting the network IPv4 and IPv6 machine have to be able to talk together.
So, as long as IPv6 and IPv4 computers can't talk to each other, we will be limited to the back end servers you are talking about. Until we go beyond that point, IPv6 will stay relatively useless.
Worst case, I'd need to connect to all the computers in my P2P network - a few million, or all the computers in my bot net - a few million more. And in both of those cases, the client software either is or will be smart enough to keep me connected via a proxy of some sort.
For example - To get to Google without ipv4, I don't need to connect to every one of their millions of servers via IPV6, I only need to get a quad A from my DNS and connect to their external facing load balancers via ipv6. As soon as I can do that, I don't need ipv4 to use Google. For all I care they could be running Arcnet on their back end servers.
Another consideration is the prevalence of NAT and proxies in the corporate world. Corp users are already NAT'd, Proxied, Content Filtered anyway, so upgrading the the corporate web proxy to do 4 to 6 translation puts the entire corp desktop on the 6 network in one shot, and isn't a major re-think of Corp networking.
As someone who runs such a site, I'd only think about upgrading the internal networks to v6 long after the public internet has gone to v6.
I agree, though, that exposing your services to the Internet via both 6 and 4 and leaving the backend internal data center at 4 is probably the way to go for the next decade or so. Heck - we are still running DecNet on production servers.