I have Radeon HD 5970. As root I can do startx and use everything. All has been configured correctly and Mesa OpenCL is even available. I upgraded from xenial-proposed the next three packages too see if it would fix it:

  liblightdm-gobject-1-0:amd64                1.18.3-0ubuntu1
  lightdm                                     1.18.3-0ubuntu1
  ubuntu-drivers-common                       1:0.4.22

It did not. In /var/log/syslog I found:

Sep 15 16:57:44 ### systemd[1]: lightdm.service: Service hold-off time over, scheduling restart.
Sep 15 16:57:44 ### systemd[1]: Stopped Light Display Manager.
Sep 15 16:57:44 ### systemd[1]: Starting Detect the available GPUs and deal with any system changes...
Sep 15 16:57:44 ### gpu-manager[1950]: /etc/modprobe.d is not a file
Sep 15 16:57:44 ### gpu-manager[1950]: message repeated 4 times: [ /etc/modprobe.d is not a file]
Sep 15 16:57:44 ### gpu-manager[1950]: Error: can't open /lib/modules/4.4.0-38-generic/updates/dkms
Sep 15 16:57:44 ### gpu-manager[1950]: Error: can't open /lib/modules/4.4.0-38-generic/updates/dkms
Sep 15 16:57:44 ### gpu-manager[1950]: update-alternatives: error: no alternatives for x86_64-linux-gnu_gfxcore_conf
Sep 15 16:57:44 ### systemd[1]: Started Detect the available GPUs and deal with any system changes.
Sep 15 16:57:44 ### systemd[1]: Starting Light Display Manager...
Sep 15 16:57:44 ### systemd[1]: Started Light Display Manager.
Sep 15 16:57:44 ## systemd[1]: Started Session c10 of user lightdm.
Sep 15 16:57:45 ### systemd[1]: lightdm.service: Main process exited, code=exited, status=1/FAILURE
Sep 15 16:57:45 ### systemd[1]: lightdm.service: Unit entered failed state.
Sep 15 16:57:45 ### systemd[1]: lightdm.service: Failed with result 'exit-code'.
Sep 15 16:57:45 ### systemd[1]: lightdm.service: Service hold-off time over, scheduling restart.
Sep 15 16:57:45 ### systemd[1]: Stopped Light Display Manager.
Sep 15 16:57:45 ### systemd[1]: gpu-manager.service: Start request repeated too quickly.
Sep 15 16:57:45 ### systemd[1]: Failed to start Detect the available GPUs and deal with any system changes.
Sep 15 16:57:45 ### systemd[1]: lightdm.service: Start request repeated too quickly.
Sep 15 16:57:45 ### systemd[1]: Failed to start Light Display Manager.
Sep 15 16:57:45 ### systemd[1]: Stopped Light Display Manager.
Sep 15 16:57:47 ### systemd[1]: gpu-manager.service: Start request repeated too quickly.
Sep 15 16:57:47 ### systemd[1]: Failed to start Detect the available GPUs and deal with any system changes.
Sep 15 16:57:47 ### systemd[1]: lightdm.service: Start request repeated too quickly.
Sep 15 16:57:47 ### systemd[1]: Failed to start Light Display Manager.
Sep 15 16:59:44 ### systemd[1]: Stopping User Manager for UID 104...

And the output of gpu-manager is:

# gpu-manager 
last_boot_file: /var/lib/ubuntu-drivers-common/last_gfx_boot
new_boot_file: /var/lib/ubuntu-drivers-common/last_gfx_boot
can't access /run/u-d-c-nvidia-was-loaded file
/etc/modprobe.d is not a file
can't access /run/u-d-c-fglrx-was-loaded file
/etc/modprobe.d is not a file
/etc/modprobe.d is not a file
/etc/modprobe.d is not a file
/etc/modprobe.d is not a file
Looking for fglrx modules in /lib/modules/4.4.0-38-generic/updates/dkms
Error: can't open /lib/modules/4.4.0-38-generic/updates/dkms
Looking for nvidia modules in /lib/modules/4.4.0-38-generic/updates/dkms
Error: can't open /lib/modules/4.4.0-38-generic/updates/dkms
Is nvidia loaded? no
Was nvidia unloaded? no
Is nvidia blacklisted? no
Is fglrx loaded? no
Was fglrx unloaded? no
Is fglrx blacklisted? no
Is intel loaded? no
Is radeon loaded? yes
Is radeon blacklisted? no
Is amdgpu loaded? no
Is amdgpu blacklisted? no
Is nouveau loaded? no
Is nouveau blacklisted? no
Is fglrx kernel module available? no
Is nvidia kernel module available? no
Vendor/Device Id: 1002:689c
BusID "PCI:3@0:0:0"
Is boot vga? yes
Vendor/Device Id: 1002:689c
BusID "PCI:4@0:0:0"
Is boot vga? no
Skipping "/dev/dri/card1", driven by "radeon"
Skipping "/dev/dri/card0", driven by "radeon"
Found "/dev/dri/card1", driven by "radeon"
Number of connected outputs for /dev/dri/card1: 0
Skipping "/dev/dri/card1", driven by "radeon"
Skipping "/dev/dri/card0", driven by "radeon"
Skipping "/dev/dri/card1", driven by "radeon"
Skipping "/dev/dri/card0", driven by "radeon"
Does it require offloading? no
last cards number = 2
Has amd? yes
Has intel? no
Has nvidia? no
How many cards? 2
Has the system changed? No
main_arch_path x86_64-linux-gnu, other_arch_path i386-linux-gnu
update-alternatives: error: no alternatives for x86_64-linux-gnu_gfxcore_conf
Current alternative: /usr/lib/x86_64-linux-gnu/mesa/ld.so.conf
Current core alternative: (null)
Current egl alternative: /usr/lib/x86_64-linux-gnu/mesa-egl/ld.so.conf
Is nvidia enabled? no
Is nvidia egl enabled? no
Is fglrx enabled? no
Is mesa enabled? yes
Is mesa egl enabled? yes
Is pxpress enabled? no
Is prime enabled? no
Is prime egl enabled? no
Is nvidia available? no
Is nvidia egl available? no
Is fglrx available? no
Is fglrx-core available? no
Is mesa available? yes
Is mesa egl available? yes
Is pxpress available? no
Is prime available? no
Is prime egl available? no
AMD IGP detected
Discrete AMD card detected
Driver not enabled or not in use
Nothing to do