By Microsoft's own admission something of a temporary workaround, the Windows Insider application has gained (server-side, so no update required at your end) a new option: 'Production'. This enables you to switch your phone to mimicking a non-Insiders device in order to pick up 'firmware' updates, lower level code and drivers, as we saw recently for the Lumia 950 and 950 XL. This is all something of a temporary measure, mind you, since firmware updates are being opened up to full Insider-configured handsets in the future.

From Microsoft:

While we are getting this update experience fixed; we have a temporary workaround in place to make things easier for our Windows Insiders. Within the Windows Insider application on Windows 10 Mobile we have added a new option called ‘Production’. This will point your device back to the production update servers where you will be able to get any firmware updates that are available for your devices. Once you have the update you will then need to go back to the Windows Insider application and reselect the Fast or Slow ring. Please be advised, if you don’t reselect Fast or Slow with any build newer than 10586 your device will be in a state which may not receive any future Insider Preview builds or Cumulative Updates.   Full steps to complete this process are as follows:

1. Open the Windows Insider application on your device
2. Select “Get preview builds”
   - There will now be three options to choose from:  Insider Fast, Insider Slow, and Production
3. Select “Production”
   
   
   
   
4. Read the presented message and tap “accept” to confirm your selection
   
5. Your device may prompt to reboot.  If not, complete a manual reboot.
   
6. After the reboot, check for updates.  If available, your device will display an update to receive the newest firmware.
   - Settings > Updates & security > Phone update > (check for updates)
   - If the device shows a “no updates available” message, your device has the most recently available firmware
7. If available, complete the update by following the on-screen directions
8. Once installed, re-open the Windows Insider application
9. Select your desired update ring (Insider Fast or Insider Slow) and reboot as prompted
10. Your device now has the newest firmware available and is set to receive the next flight when launched for your selected ring

In short, a little clumsy, but there's clearly work to be done server-side by Microsoft and in the meantime at least you can check for firmware updates now without any more hassle than tapping the screen a few times and doing a couple of reboots. It sure beats having to leave the Insiders Programme and hard reset the phone...