There's a good reason test-ipv6.com doesn't have an IPv6 address. If your IPv6 is broken then you can't go to that site to test it :).
If your IPv6 is working you can't go to the site to test it unless you also have working IPv4.
There is absolutely no reason why test-ipv6.com can't have both an A record and an AAAA record.
Q: Why is this web site reachable via IPv4 only?
You're right, there are no AAAA records, intentionally. A percentage of users are unable to browse sites that are dual-stack. If the users can't connect, then they can't be told they have a problem. This is a big problem facing content providers today; of which, I work at one for my $dayjob. As such, the main test page requires IPv4 (either native or translated).
At some point, when the percentage of "broken" users has gone significantly down, I'll consider making test-ipv6.com dual-stack..