Why battlecruisers would exist without armored cruisers

Would battlecruisers have been invented without armored cruisers?

Probably, but under another name and with different national design priorities. The intended role of the predecessor ship would shape the result.

A fast, heavy cruiser-type role would still emerge in this interpretation even if the name and immediate predecessor changed. Nations that treated armored cruisers as second-line battleships, notably Germany and Japan, would produce heavier battlecruiser concepts than Britain, which also thought in terms of commerce warfare.