I believe that these hard plastic insulating foams (also Kingspan and similar) with the very fine holes, have some special gas in the bubbles when they are manufactured, and this makes them better insulators than fibreglass or expanded polystyrene. Over years this gas diffuses out and is replaced with air, so they are not as good as when new. However, they are still as good as expanded polystyrene, and still better than the same thickness of fibreglass.
So they story you were given is not accurate. old foam is still a good insulator, but not as good as new foam. Don't cut it off as it is well-fitted with no gaps, thin spots or seams like a jacket might have.
My cylinder is foam-coated, but I had a couple of red jackets that I had used on the old cylinder, and I put them on top. I also insulated all the pipes with Climaflex which is stiff plastic foam and very effective. I too find that the airing cupboard is now cold, except in winter when the pump is from driving the heating circuit. I don't mind as this means I am not wasting energy and keeps my summer gas bill very low.