If bitfunder "suddenly shuts down" the list is not available because it's on their website. Though there's always this:
https://btct.co/history-bitfunder/ (don't use IE, I don't mirror the superfluous content so it breaks)
The list also does not include any way to contact the shareholders. (Another Edit/Add: Bitfunder did in fact add this option, just found it in my account preferences! I can share my email with the issuers!)

btct.co makes shares/email/bitcoin address lists available to issuers on the site (html), via the api (json), and by emailing all issuers complete lists at 12 hour intervals.
Thanks for the backup. I would love to offer users the same for btct, except that doesn't sound possible.

No, probably not. I wasn't comfortable with publishing the BTC addresses.
Someone could start up a backup service though that issuers could use. Eg, they open an account, plug in the api endpoint (w/ api key), and the backup service could archive the lists at regular intervals. This service would be good for backing up more than just the asset lists. Regular users could use it to back up their portfolio contents, etc.
Don't really need a backup of a partial list (which is all you can get on Bitfunder) when, as an asset issuer, I can get a full list of my investors on BTC.CO any time I want.
My view is that making a partial list public does more harm than good (it gives out information that shouldn't be in the public domain - allowing correlation between various holdings and also determination of the holdings in an asset by someone buying or selling). Investors have to trust the asset issuer anyway if they want their investments honoured - so why not trust them to hold the list in the first place?
Publishing an incomplete list is the worst of both worlds - it doesn't give the information necessary to continue operations whilst giving out information to everyone that they have no right to be given.
I think the public list is complete. If I understand it correctly, there's a separate private list (which includes email addresses) which is also complete, with the exception of your email address if you have not opted into sharing it.