Margin positions are typically liquidated when they go negative. At worst, the customer owes Robinhood some leftover slippage after their margin gets called. And typically, margins are called before your account goes negative.
If your trade starts going bad they'll be aggressively unwound. They're a very well-understood model offered by tons of (profitable) brokerages out there. Robinhood is not doing anything new by offering them.
That's not how margin trading works. It is a loan, but it's collateralized against your existing holdings. Most institutions will only loan you a percentage of your existing holdings (somewhere between 30% and 50%). Additionally, if the market value of your existing holdings dips such that the total loan amount is greater than the maximum percentage loan, those assets can (in most cases) be immediately sold to pay back the loan. This is called a "margin call".
To use your example, if you were using margin from an institution that allows for a 50% rate, in order to get that $10k loan, you'd have to already have $20k available in a relatively liquid assets (stocks, bonds, ETFs, mutual funds). If the value of your holdings ever dipped such that you held less than $20k, thus making your loan greater than the institutions 50% limit, some of those assets would be sold to pay down the loan.
There are additional safety measures built into margin accounts by brokers that offer them such as ensuring that the collateral assets aren't 100% allocated a single risk prone stock for example. Margin is a very well understood and safe business model for many brokers.
My understanding of the criticism of the GGP is not that Robinhood's business model is unsafe for Robinhood, it's that it's unsafe for their target market (casual stock traders). The idea is that casual trading is, for most of the population, gambling. Depending on your moral stance on gambling, using "free trades" as a marketing funnel for loans designed to be used for gambling falls somewhere between grey and repugnant.
Isn't a credit check required to open an account?
I think you're missing what the criticism here is.
Let's say you, right now, have $10,000. I say "Hey, lend me that money so I can play the stock market with it." Would you give me the money? You know literally nothing about me. You don't know if I've ever traded a stock before. You don't know if I can ever pay you back. There's a higher than normal probability that I will never pay you that money back.
Robinhood is taking on a lot of risk and using that as proof that they're a profitable company.