When you submit an application, you will usually be told what to include. Follow those rules first. If there is room to submit other materials then, providing your manuscript seems like a good idea. The usefulness of providing the manuscript will of course vary depending on the role of the position you seek. Posting the manuscript somewhere can be useful for several reasons but when applying for a post, people evaluating an application (one of probably many) will not want to have to spend time gathering the materials themselves, they want it provided with the application (unless clearly stated otherwise).
What you need to ask yourself if the manuscript will reflect well on you? From your answer, apparently so, but a poor manuscript should probably be avoided. An assessment of the future publishing of the manuscript is not something I generally would rely on.
The person or committee evaluating applications will chose to judge your manuscript as they see fit. Usually, a written thesis or manuscript can help to assess important aspects of a person's capability. A person evaluating the application will of course also try to judge if you are the sole originator or if others have a large imprint on the work. It is therefore important to provide a good account for what is truly yours and what can be attribute to others, i.e. list the contributorship to the work. If you want to see discussions on such issues, please search the tag authorship on on contributorship here on academia.sx.