The exchange is just that.. an exchange, people use it as gateway to the world of dollars. When they're done trading users can (and ought to!) withdraw their funds to their own wallets. Like anything else that requires an extra step, many do not (or delay a long time before doing so).

> Couldn't that function be equally served by some kind of high-risk brokerage account that has a crypto investment option?

Absolutely. What coinbase does could be done by traditional brokerage accounts. This is one of the reasons that people have been critical of coinbase as a business.

The regulatory uncertainty around Bitcoin has so far mostly kept more traditional brokerages out of the space. Though, FWIW, Interactive Brokers has a partnership with PAXOS to support bitcoin on their platform, the integration is not amazing however.

It's also the case that cryptocurrency exchanges make a considerable amount of income from promoting varrious extremely sketchy alternative cryptocurrency and accepting massive bribes for their listings and other activities that traditional brokerages wouldn't likely be interested in engaging in because they're already illegal for the other assets they handle.