As someone who works with scholarly data, having IDs for things is absolutely invaluable. There's a lot you can do with DOIs quickly and easily that would be prohibitively difficult if you had to use human-added metadata.
It's good to hear that scholars are getting together and trying to mitigate the volatility of URLs. I'll definitely need to read more on the subject to understand the pros and cons so thank you for your introduction in the article; many specialist blogs tend to only write for their specific audience. It seems like DOIs are an ample solution as long as they aren't a single point of failure.
Here's a short list of posts that will give some background:
http://crosstech.crossref.org/2015/06/rehashing-pids-without...
http://crosstech.crossref.org/2015/03/january-2015-doi-outag...
http://crosstech.crossref.org/2013/09/dois-unambiguously-and...
http://crosstech.crossref.org/2010/03/dois_and_linked_data_s...
A DOI is a Digital Object Identifier.
A digital object is something like a scholarly article. People need to be able to find it by URL, cite it, and refer to it with an ID. The URL is no good because it could change. If you cite a URL and it 404s in a few years time, that's link rot.
So a DOI is a link that points to the item. It 'belongs' to the publisher who owns the article. They are obliged to update the link to point to the new location when it moves. Hence the DOI link always (in theory) works.
From a user's perspective, a DOI is just a URL that you visit that redirects you to a different URL. From inside, it's a link resolver system backed by a database that publishers keep updated through a registration agency (e.g. Crossref).
Different Registration Agencies do different things. Crossref has a tonne of metadata, contributed by publishers. Have a play here: http://api.crossref.org/works
There's a good Wikipedia article: https://en.wikipedia.org/wiki/Digital_object_identifier
Scholarly publishing is an area where documents need to stick around for a long time, be able to be cited in a well-understood way, etc etc.
Crossref is an trade association of scholarly publishers, formed to solve exactly this problem. Part of the deal when a publisher joins Crossref is an obligation to keep the links up to date and a plan for what happens if the publisher suddenly vanishes into thin air.
Initiatives like CLOCKSS address the archiving question http://www.clockss.org/clockss/Home . The CLOCKSS homepage has a good overview.
And to the 'tracking stuff down' point, the DOI is resolvable (i.e. a link you can click) but it's also an ID that you can look up in a database. DOIs predate the web and may live longer than it. Having a single, 'official' ID (whatever it looks like) is better than searching by metadata.