"But it (mostly) doesn't matter which widget library "
No, actually it does matter because otherwise you get each application with a different look & feel, with controls that doesn't behave the same in subtle ways. Interaction among different application and the OS becomes much more cumbersome.
"it will bring in the necessary libraries as part of its installation"
That another Linux issue that brings compatibility problems when the "necessary libraries" change their ABI/API at will - and each distro may be only compatible with specific versions. Moreover if they are not OS standard libraries you start to enter the GPL hell for non-open source applications.
"UI widgets might look slightly different in different apps"
It is a problem especially when they **work** slightly different and or developers have to try tricks to make them work the same way, or users will be baffled why something that work in a way in an app works differently in another.
"certainly Windows has previously been (possibly still is) guilty of having programs using all kinds of wacky interface libraries"
True because they are aimed at the kind of users who likes bells & whistles, like multimedia players. But for example nVidia much more standard UI Control Panel was much simpler to use than AMD ones.
As long as Linux does its best to keep desktop application developers away, its user base will be small. Still it can thank Microsoft which is quite busy trying to alienate its desktop users too....