Because 120 was an open standard. By pushing 620, they ensured that your camera would always have rolls labeled "Kodak" in them.
And this worked -- until GAF and Agfa film started to show up in 620. There was never (could not be) a patent or trademark on the 620 spool and the film was identical to 120 except for a few digits in the print on the box and backing. So when other companies decided Kodak was selling enough 620 film to notice, they jumped into the stream too.
did Kodak only use metal spools for 620?
As far as I know, the first time 620 spools appeared in any other material than brass or steel was when Film Photography Project needed to produce them and (presumably for cost reasons) selected plastic for their respooled products (though B&H and Freestyle have sold respooled film for years, I think they always used metal spools, possibly recovered ones from photo labs -- seems to me Kodak used to buy them back for many years, and when that program stopped, there were probably thousands of labs suddenly stuck with a big box of spools in the back room). And, of course, if you developed your own film when you could still buy factory-rolled 620, you got a new spool with every roll.
Before 120 switched to plastic spools (1970s? 1980s?) all paper backed film came on spools with metal flanges, though sometimes the core was a short length of wood with the metal slipped and pinned on.That's 828, 235 (35mm with 24x36 frame and paper backing), 127, a number of oddball formats, three or four that got merged into 127 or 120, 116/616, 122, 124, and on up to 4" and 5" wide roll film for the bigger pre-WWI folding cameras. There was an entire industry within the photo industry, just making spools.
Fortunately, the old Kodak spools last almost forever (I've got one that's rusty, but I guess it must have gotten damp or been exposed to stop bath). We should still be able to shoot 620 cameras for as long as 120 film is produced.