Could you point us to examples of how Mozilla's User Advocacy Team
"distills information gained through user feedback" into something
Sure. So Input  is just one tool of the trade that UA at Mozilla uses they also use Crash-Stats, BMO, SUMO and other sources to generate User Sentiment reports  which are available to all the teams and the community at large and are one form in which feedback and user experiences are made into an actionable data.
Recently in Firefox 24.0 Mozilla was able to find that the Norton Toolbar used by Windows users was causing crashes and disappointment among users and with this early feedback which might not be noticeable in crash-stats they were able to swiftly work with Symantec to get them to create a compatible build.
I fear that a prominent feedback tool won't be of much use
without a solid plan for how the feedback would be processed. Worse,
it may even direct users away from more appropriate venues for their
issues. For instance, a crash report submitted to the error tracker
 is much more valuable than a user submitting feedback that "foo
keeps crashing." A criticism of a package in Universe would probably
be better placed as a review in the Software Center so other users
could make an informed decision when considering installing it.
I totally agree and that's why this is just a proposal and if there were consensus that such a team could benefit our users. Then we could advance to making solid plans around what UA tooling would look like in Ubuntu.
Right now I think we are reactionary in addressing issues in Ubuntu and the idea behind User Advocacy is to be proactive in addressing issues impacting users and much more responsive by delivering feedback to those who make the decisions.
Speaking frankly I think we all care about making a good product but realistically I do not think teams have the time or ability to dive into the experiences our users our facing without impacting the time spent on their already busy roadmaps. The idea is UA would be an asset to these teams by aggregating, digesting and reporting data to the teams and community so a better experience could be delivered.
I came across Firefox Input  which seems to be the dashboard for
information coming in from Mozilla's feedback tool. In the past 90
days, there have 145,274 submissions. Giving some of them a quick
skim, it's a really mixed bag. Doing analysis of open-ended questions
is no simple task. Whether or not Ubuntu decides to do something
similar, I'd be very interested in how Mozilla puts this data to use.
UA in Mozilla doesn't exactly analyze that data by hand but instead relies on tools that generate reports which they then present to teams. Notably UA in Mozilla is responsible to the users that being that their job is to advocate for user needs and issues impacting users even if it means bugging managers and developers about issues until they are addressed.