I am sure Rollason was correct but he no doubt was writing for the professional designers of massive machine-tools and marine diesels as well as of those working closer to our sizes of components.
One of the bits of lab equipment I operated at work was a vertical cylinder about 14 ins diameter with a wall a good inch thick, topped by a massive plate lid needing an overhead hoist, and floored a few inches below by a deep-skirted, cast-aluminium piston. The lid was mild steel but the cylinder, including its downwards extension forming supports etc., was cast-iron for that very reason – damping. Just to make sure, the whole caboodle was also bolted to a massive concrete plinth.
A chuck back-plate on a moderately-sized lathe is frankly pretty unlikely to resonate noticeably. Yes, if you dangled it from a bit of string it might make a nice if perhaps high-pitched dinner-gong, but in service it is heavily clamped between the mandrel and chuck and would probably only ring if at all at very particular speeds or under certain cutting conditions. More likely any ringing you hear is from the work-piece.
Nevertheless even quite small machine-tools do use cast-iron for its stability and self-damping as well as the other reasons Rollason gives; but some modern ones are built around welded steel fabrications filled with special forms of concrete. I recall reading this of the products of a major German builder of very large machining-centres; but I think the Peatol lathe successfully uses much the same principle.
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Having typed that about gongs, I recall once being a passenger on a freight ferry from England to Sweden. The steward's dinner-gong was a steel pipe blanking-plate suspended from a piece of wood!