I dare say it will be a while before IPv6 only sites appear.
They are here already, some of them have been here for years. Ofcourse noone in the so called "real world" care about them, since the "real world" has only been IPv4.
Till now.
As mentioned, next year there will be no more new IPv4 addresses available, meaning that new allocations will come from dealing with IPv4 addresses already owned by people. This means IPv4 address blocks might end up as a commodity you have to pay for. In addition, as IPv4 blocks will shift around here and there, the reverse delegations in DNS will become more complex and the routing tables will also become more and more complex as more and more smaller IPv4 segmens becomes spread around. All in all - lots of work and money, expences that are easily avoided by simply shifting to IPv6, a protocol already supported by a large majority of the users and the equipment in between.
There's too much in the way of entrenched IPv4 stuff out there in the real world for a simple "switch" from one to the other.
Well, to some extent that is true, we will probably never really get rid of IPv4 - however, once IPv6 is established as "something most users have", you can bet your ass IPv4 services will start to vanish - rapidly, as there is work involved in maintaining both IPv6 and IPv4 in parallelle.
Wow, how did we get onto this again? :lol:
Expect this to come up more and more frequent in the coming years.
Oh - and no magic router will help you, sorry.