A couple of points:
It may well be that you can't train for a night rating at an unlicenced airfield, but you can certainly use an unlicenced airfield for t/o and landing at night. I certainly do. It would be rather restrictive otherwise as some licenced airfields become unlicenced at night because it would be uneconomic to staff them all night.
I'm not sure about the insurance being limited to licenced airfields. I have never had an aircraft policy that restricted operation to licenced airfields and I have never been asked the question when filling in proposal forms. I don't know anyone who has their aircraft insurance limited to licenced airfields and I wonder if this is just used by some clubs as an excuse to say to their hirers that they don't want them to use unlicensed airfields. For example, I have been refused permission by one club to use ANY unlicenced airfield in their aircraft, even in a Cessna 152.
If anyone here actually has an aircraft policy that prohibits flying into unlicenced airfields I would appreciate them telling me the exact restriction and which insurance company they are with. I will then stand corrected, but at present I suspect it might just be a good story.