Is there any way to provide a menu link for people to view the most recent and/or all past Newsletters on the website or is it only designed to be e-mailed and you have to essentially go into Pages and create a Newsletter page and copy the content?
You can have a site news section (Content > Site news). The site news can be placed in a block, of course, and placed on a page.
IMHO, I don’t think a news letter is a good way to display site news. News letters make much better email marketing tools.