As others may have already mentioned, if MTP is working properly, your sdcard will be recognized by your PC just like your internal storage is. No, it's not mass storage but you should still be able to drag and drop any type of file on your sdcard. You might get a message saying that the file type isn't compatible (or something like that), but you can still save it on your sdcard. Btw, this is also true for internal storage. You should be able to drag and drop ANY file type on your internal storage. So if you can't drag and drop, that's not a fault of MTP, it's an issue with either your phone, sdcard, PC, drivers, operating system or something else.
I also don't know why you're so hesitant in doing a factory data reset. Yes, it's a pain to have to set up your phone, download all your apps and redo all your preferences. But if you're using a wifi connection and don't have a gazillion apps, the whole process will likely only take two hours as the most. Besides, it already sounds like you've spent way more time than that trying to figure out your issues to no avail. At this point, that appears to be your only option.