I was also against not doing native controls for quite long time, mainly because I thought using native controls was the only way to give the system (Windows for example) ability to read the text and other data (screen readers, automation, etc.)
It turns out that's not the case. There is an ACCESSIBILITY api which Qt implements that allows the OS (Windows) to "read" text and control the GUI in the non-native widgets.
This, and the fact that it's much more simpler to do things by subclassing the C++ way.
I also have wxWidgets experience, and here is my summary: Lots of Leaky Abstraction. For example - try to get the current line of a text-edit control - internally what wxWidgets does (or was doing time ago) was to send message to the control, but if the control did not support the message, it had to reparse the whole text, and based on your byte-position count how many CR-LF were there and give that information to you.
e.g. it was hard to predict which operation is going to take this and that much of time.
Qt is not without pitfalls, and missing features - Visual Studio Docking like windows for example that supports multiple monitors, but other than that it was much easier for our team to use it, rather than MFC or the Win32 API, and even wxWidgets.