Having a working personal Apple ID should help facilitate reaching Apple Support quicker, but you don’t need a working Apple ID to use the support site, or to receive support over the phone or in person at an Apple Store.
Even quicker still may be using the iOS Apple Support app if you have an iOS device handy. Doesn’t even have to be yours, you could borrow a friend’s. In my experience, setting an appointment on the support site is very quick and easy with callbacks under a minute being normal for me.