OK, so after reading a lot, I have figured out why this is a problem, and I do not need to use any workarounds.
The problem seems to be (at least in my case), that since the place where you enter text is hidden initially (or nested or something), AlertDialog is automatically setting the flag WindowManager.LayoutParams.FLAG_ALT_FOCUSABLE_IM (or some combination of that and WindowManager.LayoutParams.FLAG_NOT_FOCUSABLE) so that things don’t trigger a soft input to show up.
The way that I’ve found to fix this is to add the following line after the dialog has been created:
dialog.getWindow().clearFlags(WindowManager.LayoutParams.FLAG_NOT_FOCUSABLE|WindowManager.LayoutParams.FLAG_ALT_FOCUSABLE_IM);
Once this is done, the EditText acts like a normal EditText, no kludges or workarounds necessary.