It's a CofA aeroplane.
If the engine has gone "on condition", the only person able to learn on it is the sole owner or members of their immediate family.
If it is not yet on condition, anybody can learn, or be tested on it.
Unless the instructor does it for nothing, in which case it's all fine. Historical reasons.
Stuff about insurance, RTO, etc. is all true.
The stuff about "on condition" may change - the current proposals for ANO amendments suggest significant relaxation of the ownership rules for instruction. However, that's 1-2 years away, and the final form not yet known.
G