The Wayback Machine doesn't offer a way to submit an entire site, only a single page as you've already found. This is touched on in a couple of points of their Wayback Machine FAQ:
Can I add pages to the Wayback Machine?
On https://archive.org/web you can use the "Save Page Now" feature to save a specific page one time. This does not currently add the URL to any future crawls nor does it save more than that one page. It does not save multiple pages, directories or entire sites.
and
How can I get my site included in the Wayback Machine?
Much of our archived web data comes from our own crawls or from Alexa Internet's crawls. Neither organization has a "crawl my site now!" submission process. Internet Archive's crawls tend to find sites that are well linked from other sites. The best way to ensure that we find your web site is to make sure it is included in online directories and that similar/related sites link to you.