Earlier quoted context omitted.
Doesn't that rely on you being the actual owner of the shares? If your shares only exist in Robinhood's database, I don't think SIPC would apply. EDIT: After some Googling it appears that SIPC does protect securities that weren't actually purchased due to fraud or mismanagement. IANAL so I don't know if that would apply to this specific situation, but they have protected other non-owners before like in the case of Be…
Even with full shares they are held in street name - that is the brokerages name. So the story isn't much different than it is today.
A broker which just has pretend stock ownership as a book entry with the broker is called a "bucket shop".[1] A crime in most US states since the 1920s.
[1] https://en.wikipedia.org/wiki/Bucket_shop_(stock_market)