Wool and weaving led naturally to dyeing, and in Coventry this was a highly specialised craft. This led to the development of a blue dye created from sloe berries and other ingredients, that gave a rich, fast colour. This dye led to the expression 'As true as Coventry Blue', for the dye survived washing. The manufacture of this dye was a closely guarded secret of the guild in Coventry and was passed by word of mouth, from master to apprentice. So successful were they in protecting their recipe that today nobody knows how it was made, or even what colour it was exactly.
There's nothing wrong with a rebrand so long as there's a charismatic figure prepared to lead the marketing for it. Jimmy Hill did that for the Sky Blue Revolution, and fans understood exactly what he was trying to do. But you try to slip it in quietly, like McGinnity did with the badge, and all you'll do is get resentment.