After many hours of google-looping searches and several restored vCenter appliances, I found a fix (that works for me at least).
Credit to Shrikant @ https://virtuallyvtrue.com/2019/07/10/post-upgrading-to-vcenter-server-appliance-6-7-update-1-update...
First step is to spin up a new temp vCenter server, on a different IP/FQDN, to the exact same version/build as your broken vCenter server and follow the steps in the link or those below for step-by-step guide.
IMPORTANT: Take a backup of your 'broken' vCenter Server via the VAMI before starting, just in case! 
Then, on the New Temp vCenter Server:
Enable SSH
Connect via SSH (Use PuTTY or similar) as root
Enable the shell
Execute the command: chsh -s /bin/bash
Connect via SCP (Use WinSCP or similar) as root
Browse to the following directory: /etc/vmware/vsphere-ui/cm-service-packages/com.vmware.cis.vsphereclient.plugin
Copy the 3 sub-directories below (your version numbers may differ as these are from a v6.7u3 VCSA) to a temp location on your local PC.
com.vmware.fallback.mixed-6.5.0
com.vmware.vcIntegrity.vcIntegrity-6.7.0.41905
com.vmware.vum.client-6.7.0.41905
Next, on your 'broken' vCenter Server:
Enable SSH
Connect via SSH as root
Enable the shell
Execute the command: chsh -s /bin/bash
Connect via SCP as root
Browse to the following directory: /etc/vmware/vsphere-ui/cm-service-packages/com.vmware.cis.vsphereclient.plugin
Rename the 3 existing directories to .bak (Again, your versions numbers may differ. There may be additional directories - do not touch these)
com.vmware.fallback.mixed-6.5.0.bak
com.vmware.vcIntegrity.vcIntegrity-6.7.0.41703.bak
com.vmware.vum.client-6.7.0.41703.bak
Copy the 3 directories from the temp location on your local PC into the same directory on the vCenter Server
Execute the command: cd /etc/vmware/vsphere-ui/cm-service-packages/com.vmware.cis.vsphereclient.plugin
Execute the command: chown vsphere-ui:users *
Execute the command: ls -l
Confirm the user:group permissions have been updated to vsphere-ui:users for each subdirectory
root@mc-vcsa-v-101 [ /etc/vmware/vsphere-ui/cm-service-packages/com.vmware.cis.vsphereclient.plugin ]# ls -l
total 28
drwxr-xr-x 3 vsphere-ui users 4096 Oct 3 23:10 com.vmware.fallback.mixed-6.5.0
drwxrwx--- 3 vsphere-ui users 4096 Oct 3 23:01 com.vmware.fallback.mixed-6.5.0.bak
drwxrwx--- 3 vsphere-ui users 4096 Oct 3 23:10 com.vmware.imagebuilder.imagebuilder-6.7
drwxrwx--- 3 vsphere-ui users 4096 Sep 28 11:11 com.vmware.vcIntegrity.vcIntegrity-6.7.0.41703.bak
drwxr-xr-x 3 vsphere-ui users 4096 Oct 3 23:10 com.vmware.vcIntegrity.vcIntegrity-6.7.0.41905
drwxrwx--- 3 vsphere-ui users 4096 Sep 28 10:49 com.vmware.vum.client-6.7.0.41703.bak
drwxr-xr-x 3 vsphere-ui users 4096 Oct 3 23:09 com.vmware.vum.client-6.7.0.41905
root@mc-vcsa-v-101 [ /etc/vmware/vsphere-ui/cm-service-packages/com.vmware.cis.vsphereclient.plugin ]#
Reboot the vCenter Server.
Once it has fully booted, login and you should see the Update Manager plugin once more.
Hope this helps someone.
Martin