Online Wii games stopped working just a few years after they were released... why?
Two reasons. First, Wii online multiplayer matchmaking was through GameSpy, and GameSpy went out of business a little over a year after the Wii U was released. Second, GameSpy's library was linked statically into each Wii game, and Nintendo neglected to include a patching mechanism for disc games in IOS, the operating system of the Wii. Nor could it issue an update to IOS to allow this because of the Wii's anemic (0.5 GB) internal storage. (The same lack of space is why downloadable "channels" (applications) were limited to about 40 MB.) So even if it wanted to issue a patch to allow use of third-party game servers, it had no official means to do so.
I even miss the little world weather thing. I'd often pop in to just twirl the globe for a moment... it couldn't possibly cost anything significant to run the freakin' weather server, could it?
Yes. Licensing current conditions and forecasts from a commercial service with worldwide coverage costs money. Some national government agencies, such as the National Weather Service in the United States, provide without charge, but those are specific to each country, and integrating all countries' weather information also costs engineering time=money.