Wikipedia says the coat of arms badge was worn from 1965-1977. Whether the new (current) badge was generally well-received, I've no idea. It must've seemed like a radical redesign at the time.
From "Beautiful Histories"

In the Fifties and Sixties a roundel badge displaying the shield of the council’s arms surrounded by the club’s name became popularly linked with ‘The Blades’. The sheaves in the green field clearly refer to the name of the town and the river Sheaf and the arrows stand for cutlery in this heart of the steel industry. This emblem appeared on trade cards, series of chewing gum wrappers, postage stamps and in opponent’s match programmes. Although recognised widely as representing Sheffield United, the club did not use it until the 1975-6 season.

On prestigious occasions, such as the Cup Finals of 1925 and 1936, Sheffield United wore shirts which carried the complete coat of arms by courtesy of the council, including the supporting statues of Thor with his hammer, and of Vulcan with pincers, sitting on an anvil, obviously alluding to the city’s principal industry. The lion in the crest represents the Talbots, Earls of Shrewsbury, who once held the manor. True Blades should be able to tell you if the Latin motto ‘Deo adjuvante labor proficit’ truly meant ‘By God’s help labour succeeds’ on those occasions….

In the mid 1950’s two crossed blades with the white rose of Yorkshire, set in a black shield, appeared on the player’s blazers. The present roundel design was established as an official logo in the 70’s. In fact, the club were one of the last to have a copyright mark and the present badge didn’t appear on official company reports until 1981.
The crossed blades in the fifties is said to have been thought up by Jimmy Hagan.