> lose styling consistency
Yes. And that is (almost always¹) a good thing.
Only the designers of an app, product-owners etc. want their app to "look consistent over platforms".
Your users want the file-dialog, window-chrome, menus etc consistent too. But for them consistent means consistent with the 20+ other applications they use on a daily base. So native.
¹ Obviously some software excepted. E.g. categories like "expert software" like Blender, AutoCAD, Photoshop/CS, where dialogs must a) be optimized for their niche workflow and b) remain consistent for that user when they upgrade their OS or move between OSes. But that's an exception. Your TODO-list app or PDF reader almost certainly is not that.