CC company is jointly liable with the supplier to ensure you get the goods. That has been done. If someone at the card company is advising otherwise, I’d want it in writing and I’d bet they’d not be willing to give you that. Also redundant if you’ve used a different card company anyway.
Some credit cards (don’t know if Tesco is one of these) give additional “purchase protection insurance” or similar, which covers you if something you’ve bought using their card is lost/damaged/stolen within 30/60/90/whatever days of you buying it. Could that be what is being referred to in this instance? If that’s the case, it might be pretty handy for this sort of thing, but again I’d want to be reading the small print to make sure I was covered.
I think paying the £60 to insure with your courier is the best option, and what I’d do. Whether you should have to pay them extra to insure against them failing to do their job properly, is a whole different discussion…