Most Chinese speakers can understand (written) Mandarin, but all the spoken languages can be written in theory. Mandarin was just the first one to have been popularly written and it was the official dialect, so it became the de-facto standard (before it became de jure).

People in Hong Kong for example have been writing Cantonese for decades, albeit more often recently than before. But if you looked you'd find published works written in Cantonese 100+ years ago. That said, if you only knew Mandarin it will probably only take you a couple days to get the hang of reading Cantonese since it's not that different when written.