Snippet from the log:
Feb 16 09:23:34 rdnssd[585]: Get IPv6 address & DNS from RA
Feb 16 09:33:25 rdnssd[585]: Get IPv6 address & DNS from RA
Feb 16 09:43:17 rdnssd[585]: Get IPv6 address & DNS from RA
Feb 16 09:53:11 rdnssd[585]: Get IPv6 address & DNS from RA

The DNS is not the main issue (as I mentioned above, my ISP resolves IPv6 addresses when I use its default IPv4 DNS server).

The main issue is that dhcp6c-state is not called: I do get an IPv6 address (without DNS), but radvd is not started and no route is advertised to my local network. Therefore none of the devices on my local network will have an IPv6 address.
Using telnet to my router I noticed it had an IPv6 address and I could use ping6 to verify it worked. After manually editing dhcp6c.conf to call /sbin/dhcp6c-state and killing/starting dhcp6c my local network had IPv6 connectivity.