I haven't watched the video yet as I'm at work, and I will probably get shot for saying it... but there seems to be a bit of an exaggeration going on here...
These, as some of you seem to know, are common here in Brazil. I shower twice a day using one similar to this, 220V nominal voltage. It's actually a "fancier" one with an electronic heat control, but overall same concept.
1) I never felt any tingling. Maybe some of you are super sensitive...
2) If properly earthed, I don't see how it would be more dangerous than any other equipment. And you know what (now I will get shot), even without earthing it's not like end of the world... it's not that low of an impedance between a shower sprinkling water 30 cm above you and ground for you to certainly get deadly zapped if a wire goes loose inside. Again, of course it should be properly earthed just like anything else, but a metal chassis of any other equipment (say, a fridge) could potentially be much more dangerous may a wire come loose inside it and the chassis is not earthed
3) There are (or used to be) some with an isolated heater element - but they were less efficient and I think might have been discontinued
It was mentioned below about the fact that the earth wire is made of copper and it would get crusty / corroded and not perform - that may be a problem indeed but I think most I have seem are actually aluminum or some other metal less bound to cause problems. The ones that seemed to be copper did show corrosion definitely, but I remember measuring its resistance once as I had the same thought and it was not noticeably higher (I made very light contact with the side of the probe to make sure I was not poking through the corrosion).
I tried to find some info about deaths linked to these showers and couldn't find anything - that doesn't prove it's safe of course, but the fact these are broadly used here in Brazil and it doesn't seem to be that bad... corroborate my opinion it's not such a suicidal thing to shower in one of these. There are much easier ways to get electrocuted.
So bottom line - if properly installed, I don't think these are any more dangerous than anything else really.