Anyone acting as an employee should be covered by the employer's insurance and provided that person acts with due diligence and within the regulations and rules that should be sufficient to provide adequate professional indemnity.
Here's the crunch!
How can you say that you acted within the rules as laid down when your employer hasn't got any documented rules? Do you know for certain that your employer has public liability insurance?
The other issue concerns those instructors working as 'self-employed' instructors (which in itself is another discussion). These instructors I would agree should definitely have some form of PII.
Not advertising their services but I have found On Risk at Southampton to be very helpful in the past and I think I will be speaking to them again pdq.