I get the Idea of how brokers work, but an Exchange dealing with Bitcoin is simply acting an Escrow while brokers have a different business model than a Bitcoin exchange.
Stock exchanges are not designed for everyone, but Bitcoin is designed to be widely used for normal users who wont go through the hassle of opening an account and linking it to an exchange just to buy a bitcoin.
coinbase operates the same way as you described above, but coinbase without an exchanges like Bitstamp wont be possible, unless the adoption of Bitcoin by merchants and their daily transaction volume becomes equal to the need of users to buy/sell.
Believe me if I tell you that I did a thorough due diligence on the matter and I've received formal replies from financial regulators on more than one EU country: Bitstamp is absolutely non compliant under EU regulations, I can guarantee you that.
Coinbase is a completely different business: they are not an "exchange", they are a "
bureau de change" (like the one in which you exchange currency at the airport): the main difference is that when you are buying or selling on Coinbase your legal counterpart is Coinbase itself, and NOT another customer like it happens on Bitstamp, Kraken, etc.. When you are sending money to Coinbase from a legal POV you are transacting with Coinbase itself, they are not holding the money on your behalf while you transact with another customer - the reason they are operating that way and NOT as an exchange of the kind of Bitstamp (which is much more profitable revenue wise) is PRECISELY because they are a serious company and they know they cannot do that - at least currently, as having a banking license (as it IS required for a trading exchangr as per Bitstamp) is not only expensive but very lengthy as the barriers for newcomers are high.
If you read carefully the ToS of the emerging exchanges you will see hints of what I'm saying: the new one based on the UK is registered as a "bureau de change" in a Coinbase fashion precisely to avoid to adquire a banking license; the one just launched in Mexico says clearly that by sending them money you are adquiring their own virtual currency, which only purpose is to buy and sell cryptos, and that they DO NOT HOLD customer deposits (because from a legal POV you are buying their own "virtual currency" by wiring them money) - those are just tricks to try to avoid the hard cold truth, which is that you need to be a bank to hold deposits on behalf of third parties.
About the escrow: I've never looked into that business model, but I approached regulators explaining them EXACTLY how a Bitcoin exchange work and they have told me that under the PSP (payment service provider) / financial intermediaries EU regulation I needed to either have a banking license or to have a banking partner holding, controlling and supervising all customer deposits (because that money is not YOURS, it belongs to your customer). While I'm ignorant on that aspect, at this stage I'd bet that an escrow would need to comply with the same regulation, as they are acting as financial intermediaries by holding funds that do not belong to them.