While doing some work research I found this briefing from the UK House of Commons library which, I think, explains the situation around the requirement to travel on a British passport or with a certificate of entitlement more clearly than most.
These briefings are designed to support the activity of Parliament and I have always found them useful and up to date. Hopefully it I’ll help people working through this situation.
https://commonslibrary.parliament.uk/research-briefings/cbp-10344/