Front tooth root short. Can this tooth be saved? Recommended that I get an implant, but I do not want to extract the tooth.

A previous trauma might have led to this condition when the root is getting resorbed. Looking at the present condition, it would grow loose eventually. It would be better if you get it extracted and replaced with a dental implant so that it is a permanent solution for the issue.

if there is no mobility in the tooth, I don’t see why you couldn’t get a root canal and keep the tooth for many more years. 

Hello at least the tooth is moving you can save your tooth doing a root canal and do regular control every period of time.