The Money team was one of the first Microsoft Groups to realize the web was a paradigm shift. They did two things that are noteworthy.
First, they very early on integrated themselves with banks (sort of a Proto Programmable Web). Most people don't realize Money was connecting to banks a year before Quicken (who offered imports of files but no direct connect)
Second, they realized the web was also a UI shift and wrote a shrinkwrapped app as a web application (I suspect not making it a shrinkwrapped app was beyond their power). In fairness they were clearly inspired by Quicken's use of hyperlinks in program but they took what Quicken did and went one better.
You can see a lot of that from this screenshot (note it's 2002): http://images.sudhian.com/articles/ShuttleLinux/PartIV/MSMon...
MS Money was actually a revolutionary product in many ways. So much so that people didn't realize how revolutionary it was at the time.
As a s/w developer, I could immediately see that Money was suffering from years of neglect. I predicted that it would soon be cancelled and that was one of the reasons I stuck with Quicken (though it too sucks).
Due to the reasons you articulate re: Money's early web focus and my hunch about years of neglect, I think that Money's cancellation had much more to do with internal Microsoft priorities than with the rise of the web finance software companies.
For financial information, I want multi-decade records. With a web startup, I have no assurance that all the time I spend entering and categorizing stuff will pay off if the web guy goes under with my data. With a desktop program, at least I can keep going on for a long time with the old version (I'm still using Quicken 2007).
My wife and I have been recording our family finances in MS Money since January 2003. Maybe we hit a sweet spot with Money 2004 but we never saw a compelling reason to upgrade. I have an uncle who similarly is currently using Quicken from something rediculous, like 1999. The value of the records now, the ability to look back at how various changes (births, moves, Katrina, etc) affected our savings and spending has been unbelievably helpful not only in terms of making informed decisions about the future, but also preventing fights about finances.
As an aside, the other major source of friction has historically revolved around schedules, and Google Calendar is awesome for that. A very useful decision, I think, was giving each other write permission to each others schedules.