Personally speaking, I blame the customers.
They are paying for $0 trades to a very, very small trading firm with well-known trade-execution problems months / years before the GME instance. No serious trader actually trusted Robinhood, and nobody was surprised when Robinhood's trading ability was shown to be so weak in that timeframe.
There were many respectable banks with much stronger finances who were able to support the GME-rush. It was just the small guys without much $$$$ who failed, like Robinhood.
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If you did a bit of research, you would have found Interactive Brokers (for instance). I'm not a customer of IB, but they have plenty of online material for what exactly you're paying for.
And that is, trade execution. It matters, especially in times of trouble / times of risk. The stronger the bank, the better their ability to continue operations during weird times.
> The reason I dislike the analogy is because brokers aren't (typically) supposed to run out of shares to buy and sell.
Except they do. All the time when bubbles pop and other crisis form. Good luck selling stocks during the crash of (whatever). When the stock market is crashing, there's no buyers, so the price keeps dropping and dropping.
Without any buyers, you will never be matched and you'll never be able to sell until its too late. Understanding these mechanics is very important to any market participant.
"Flash crashes" with stop-loss orders are particularly lulzy. Your stock is automatically put up for a sale on a flash-crash. There's no buyers, so you sell the stock at a grossly lower price than expected (when some savvy buyer finally decides the price is low enough). That's when you're matched up. By the time you look at the stock, the "flash crash" is over, your stock is randomly sold and at a bad price.
Etc. etc. Its annoying, but these sorts of events happen all the time, and its important to remember the mechanics of buying/selling stocks at all times when trading.