Hello, this is Jim, thanks for the question.
Yes, so if you paid for the trip with your card for 5 of you, you are the claimant and in contract law, you are owed the full sum back. The others do not need to make a claim as you paid with your card. If the other people had their own insurance, they could claim back the money they paid but it doesn't sound like the case to me - you paid with your card, not them.
If you booked by credit card then you can make a claim - you can contact the card provider and ask for a refund under S.75 of the Consumer Credit Act 1974 - as they are jointly liable if you do not get the service you paid for. You can claim a full refund even if you only paid for part of the trip with the credit card as long as £100 or more was spent.
If the insurer is not willing to pay you back in full, you could make a complaint to the insurer, wait for their final response, then go to the Ombudsman (https://www.financial-ombudsman.org.uk/consumers/how-to-complain) though this will take several weeks if not months. Bear in mind if a flight cancels, you are due a refund from the airline - as per the European Commissions's recent announcement. It is the same with a hotel too.
If the insurer insists the other people need to claim, it may be quicker than going through the complaints procedure - but to answer your question, as above - no. I disagree with the insurer.
I hope this helps, please feel free to ask any follow up questions.
Have a good evening,