that 'asian toy' would have to be locked into using an IPv6-only website. but there's no good reason to have such a site.
yes the freely allocated IP addresses will run out, but that's not nearly the same thing as saying that all those IP addresses are going to be in use.
there are a lot of companies that have public IP addresses that aren't using all of them (and some older companies who got involved early who are using all of them, but would be happy to restructure their networks to not use them if they could sell them off)
there are also a lot of options out there to reduce the number of IP addresses in use, each with a different set of trade-offs. These include NAT (allowing end-users who have signed contracts saying that they are not going to run servers anyway to share public addresses), SNI (allowing servers running SSL to share an IP address), etc. implementing those things will free up more addresses.
yes, the ISPs may want to get companies to change the IP addresses that they are using so that they can have less fragmentation in their router tables, but that's a simple cost-benefit equation for both sides, what incentives will the ISP give companies to change their addresses (for many companies this can is probably surprisingly low, a free month of service would probably be enough)
I don't think that you are going to start seeing IPv6 only resources (other than by the IPv6 advocates) appearing anytime soon, i would be surprised if they were as close as 2015, and I would not be surprised if they were out as long as 2020 or later.
right now, the only real value that Ipv6 gives is in a private network as you don't have to contend with the limited RFC private addresses (and this can include ISPs using NAT464 to connect customers to the Internet passing over IPv6 on their internal network) for the endpoints it's really no value