Wider sentence spacing certainly predates the typewriter, but the question then was not "how many spaces" but "how much space".

Someone writing by hand obviously does not count a discrete number of spaces, and a cold-metal typesetter has a wide variety of spaces available: they might use a one-and-a-half-en space (either a single piece of type or an en space followed by a half-en space) or an em space.

It must have been the introduction of the typewriter that brought the idea of "two spaces" instead of "wider space".