You're correct that traditional content is routed through a ton of different companies, and that anyone of them can fail, but using bit.ly is essentially pointless, as it just tacts on another service that must be relied on.
In 10 years when bit.ly is completely dead, 2/3rds of the tweets or other comments on the web will be rendered useless.
People complain about this a lot, but the rate of link-rot of ordinary URLs is much, much greater than people believe. I run a URL shortener, and I can assure you that while all of our links still work, huge numbers of the pages they point to no longer exist.
The problem is also overstated because nearly all URL shorteners (bit.ly included) are sending HTTP 301 responses. This means that the short link is never indexed; only the original link is. So shorteners have no effect on pagerank/search results, which is where the longevity of pages is important (since anything not published in the very recent past is mostly discovered by search).
The project definitely needs more volunteers. Check out the website or join #archiveteam on EFNet if you are interested in helping save the future internet.
In 10 years when bit.ly is completely dead, 2/3rds of the tweets or other comments on the web will be rendered useless.