I don't know why people still claim that. Doubling up WAS a feature in the 70's and 80's - actually from the time the Second Division started.
Many BL teams adopted a second-tier team to provide riders. Riders like Eric Broadbelt, Ken Eyre, and Steve Waplington rode for both Belle Vue teams, and Gary Peterson appeared for Nelson and Newcastle. Terry Shearer turned out for Doncaster and Newport, while Peter Murray featured for Canterbury and Wimbledon. Geoff Ambrose was at Crayford and Wolverhampton.
Later on, Hull and Newcastle were linked (same promotion, I know). Colin Richardson was a staple at both Plough Lane and Arlington, as was Andy Grahame at Birmingham and Milton Keynes. No, it wasn't called "doubling up", but it certainly wasn't uncommon.