Why are some 2020 Democratic candidates insisting on outlawing private insurance
At the Democratic primary debates, the moderators asked for a show of hands for candidates that wanted to move to fully public health insurance, apparently eliminating private insurance.
That's not at all what was asked. The question, per the transcript was: "Who here would abolish their private health insurance in favor of a government-run plan?" Firstly, this is definitely not: "who wants to outlaw private insurance?" And secondly, keep in mind that the hosts only asked questions that purposely seemed to be as divisive as they could get away with, in the hopes of surfacing where the candidates disagree rather than agreed on.
The point is they'd become useless, rather than outlawed.
I should add in passing: this doesn't mean they'd be pointless. Where I live you get free healthcare -- no credit card needed. But if you've an emergency you'd better have private insurance because the waiting list is months long if not. Speaking personally I find this deeply offensive. But I could totally imagine the US ending up with such a system as a compromise.
As far as I know, many other countries with public health insurance also have a private insurance industry.
I live in one of those countries. The only thing they're happy with is that, as bad as it is, how it works isn't as screwed up as things are in the US or developing countries. FWIW I'd trade the system I pay into for some of the plans being promoted by some Democrat candidates in a heartbeat.
What is the advantage of proposing the elimination of private insurance, given that it seems to attract criticism to the proposal?
Private health insurance is parasitical in nature. There's no such thing as health insurance. Insurance is about risk. Bad health is a not a risk: it's a certainty at one point or another in your life -- and indeed, many points.
In contrast with events like your house burning down or your life ending, health problems are just commonplace. As such health "insurances" fall in between two extremes: a sort of non-profit cooperative at best (France has plenty of examples of those), and a for-profit leech at worse (see the US for examples).
Edit, seeing that the question got edited:
"When Warren and Kamala Harris raised their hands and said that they would eliminate employer-based health insurance, they made the most important gesture of the campaign so far."
Yeah... by advocating to abolish modern forms of slavery for all practical intents. No one should be losing their health insurance when they lose their job or change jobs.