EDIT : well, confirmed by skarnet, the database need to be recompiled because the fdholder interface has changed
i think the main problem do not come from s6-rc but connman update.(not sure about that for the moment). anyway remake the database should resolv the trouble for s6-rc if exist.
So first, you can compile again your Graphics database. i will show you how to do it, make the following on console (log out for your X session)
first switch to the default database
$ sudo s6opts switch Default
then erase the compiled Graphics database and the source too
$ sudo s6opts remove_db Graphics
$ sudo s6opts remove graphics
then make again the database
first add the necessary daemon on the source
$ sudo s6opts add "bundle-Connman bundle-Wpa-supplicant bundle-Dbus"
then compile and switch to the fresh compiled database
$ sudo s6opts all Graphics graphics
normaly you have now a fresh database and the current database point to Graphics, you can check it by s6opts list
now, about connman, a bug exist on it from a little while now and the change is not made by the upstream
go to the file /etc/dbus-1/system.d/connman-nmcompat.conf and change the pattern <deny by <allow. Without this a simple user can use connman-gtk
be careful on your file resolv.conf and be sure that nameserver point to the right place.
to check if connman point the address of your connection is correct, you can use this command :
$ ip addr
EDIT : maybe you can try to compile again the database for the boot time as following
$ sudo rm -r /etc/s6/compiled/default
$ sudo s6-rc-compile /etc/s6/compiled/default /etc/s6/boot-rc-serv
be sure that the symlink named current in /etc/s6/compiled point to /etc/s6/compiled/default