4th update: After a mixed bag of getting through and not getting through and getting through with poor reception, we made a call just now around 12:39 am PST to a doctor which, most unfortunately, had the same problem that I called about either early this year and/or last year...
Namely, this time, after about six minutes of conversation, we started hearing the other party's beginning of the conversation again, replayed from the start. It's the weirdest thing. So our current conversation was cut off, but we heard a recording of the other side replayed from the beginning of the phone call. :-O
I'm really sorry that that problem was never diagnosed, solved, and fixed.
Btw, just throwing it out there, but we *maybe* suspect that connection problems are worse when trying to connect to landlines in Japan. We are not certain about this, but wanted to pass it along in case it might be of help.