Here's an article that references the original research on baby hearing the ultrasound. http://www.acfnewsource.org/science/ultra_hearing_fetus.html
Also, doppler is not "Stronger" than ultrasound. They are quite different types of waves, but in general, the doppler is less strong: http://www.naturalchildbirth.org/natural/resources/interventions/interventions27.htm
Getting a fetoscope isn't really going to help at all in reassuring an anxiety prone mother. You can't hear til at least 16 weeks, and more likely 28 before reliably hearing with the fetoscope. Most women will be reassured by movements long before then. Also, using a fetoscope is a learned skill. It's not the easiest of skills, and some babies are very difficult to hear even to a trained listener.
I think reducing anxiety is a good thing in any mother. A doppler will not reduce anxiety all by itself. You can't listen 24-7, after all. It only tells you about the baby during the time you listen. The anxious mother will need to use lots of techniques to make it through the tough hours and days. If a doppler rental/purchase is part of a comprehensive plan, it seems totally reasonable to me.
Professionally, as a midwife, I've always used ultrasound and doppler as tools. Personally, with my first pregnancy, I was probably as you expect a midwife to be -
"Who needs an ultrasound? I trust this process. I want my baby, and it might be harmful, even though there isn't a speck of evidence that it is. Why risk it - I don't have any indications?"
As you all know, sometimes pride goes before a Big Fall. I miscarried. Then I had an ectopic pregnancy - something that could have killed me. Then a friend had a baby with a non-fatal birth defect discovered through ultrasound, but the baby needed to be born in hospital, and she's considered maybe no-ultrasound. Then I had two more miscarriages.
So, I've come to a place where I "need" an ultrasound so early in development (to detect a possible ectopic) and I "need" ultrasounds to reassure myself.
I'll probably doppler once a week until I feel movement. Who knows? I keep it open.