Why don't brokers take money today for pending buy orders?

`...every stock trade involves an extension of credit.

That's not true. For a cash account, the cash must be in the account in order to make the trade. There is no extension of credit since you're paying the full amount. T+2 is when the two brokers swap the cash for the security. How could Robinhood take money from your account immediately without giving you your security in return? In reality, once you make the trade, your cash is then tied up (making an additional purchase without available cash would be an account violation).

Extension of credit only occurs in a margin account.

It was in the news recently that RobinHood had to prohibit some customer orders due to a lack of liquidity. The explanation was that because brokers like RobinHood take orders today which are fulfilled after some delay, they are at risk if the investor refuses to pay, and need to enough cash in reserve to cover that risk.

Customers can't refuse to pay. As stated above, a cash account must have the full amount of cash in it to make a purchase. In margin accounts, one must have 50% of the trade's dollar amount in order to buy on margin or short (unless the broker requires more). However, the margined position can destroy account equity in which case there is insufficient account cash or marginable securities to cover the loss.

The short answer is that Robinhood is undercapitalized and had to implement restrictions in order to continue operate. Those restrictions unfairly affected some Robinhood traders and is just another reason why people should trade elsewhere.

Especially for day traders with a margin account - which is probably the majority of the traders in this context - the money wouldn’t be there yet to take, because it comes from another sale.
That previous sale will settle before or same day, but it’s not yet there yet, and with the high activity, there is some risk of parts being missing when needed: If a trader’s account runs into negative, he gets an immediate margin call, but maybe he isn’t willing or able to produce the additional cash right that day. As a result, the brokerage ends up holding the bag, and will have to pay up (until they successfully go after the trader and get their money back - which can take months, if ever). Therefore, they are required to show a certain cash to be able to cover such cases.

If you wonder how an account can run into negative: if you sold for example naked options (fulfilling the margin requirement), and the underlying does a ‘GameStop’, you could end up with a hundred times the loss of your cost basis, far exceeding all your cash and long positions.

Why does RobinHood not simply take payment when an order is placed, and hold it for the 2 business days until the order is completed? This seems like it would reduce their risk, remove the scaling bottleneck of needing to find cash on short notice, and give them a nice buffer of cash.

You misunderstand something very important. The obligation to pay for the stock within two days is not the customer's obligation. If it was, then a customer failing to make payment could cause another broker (or their customer) to lose money. It is Robinhood's obligation to make payment to the seller's broker when their customers buy securities. It is illegal for a broker to use customer funds to settle their own obligations.

Robinhood has to have enough of their own money on deposit to cover any risk that they might fail to meet their obligations to pay the seller for stocks their customers buy. This is to protect other brokers and their customers from Robinhood's failure to meet Robinhood's obligations.

If the money isn't owned by Robinhood free and clear, it can't be used to settle any payments Robinhood might need to make to sellers and their brokers. And Robinhood definitely can't just mix their money and their customer's money or use commingled funds to settle their own obligations.

I understand how that could cause problems for brokers, but I don't understand why a single broker can't unilaterally fix it. Why does RobinHood not simply take payment when an order is placed, and hold it for the 2 business days until the order is completed? This seems like it would reduce their risk, remove the scaling bottleneck of needing to find cash on short notice, and give them a nice buffer of cash.

That's exactly what they do. The problem is that other brokers don't trust them. The problem isn't their risk, it's everyone else's risk. The broker has to execute the buy pretty much instantly, and they can't take the customer's money until after the order is placed. At that point, they can't teleport the money instantaneously to the seller's broker.

Another issue is that Robinhood doesn't charge commission so to have a viable business, they have to make every penny they possibly can. Having an extra billion dollars sitting around for two days doing nothing just in case this kind of event happens is pretty much a non-starter for them. That's one of the main reasons for the two day settlement time.

For very large orders there will be an expectation of security. There may be a hold placed on a cash management account or other stock held. But it is a settlement, it could fail on either side. The NYSE has notoriously rolled back trades deemed to be obviously in error. The 2 days is for the dust to settle.

The actual cash flow can be grossed up into "gross settlement" payments in and out for the brokerages concerned.