They could, but it feels like a stretch. I suspect it would be easier to make a move for that space from an existing institutional broker like Instinet or Virtu.
Institutional business is a more lucrative market than the consumer space, and nobody can service this space at the moment, despite demand. What the market is lacking is trustworthy operators who have execution technology and balance sheet.
Coinbase have done better than other players at getting their trust brand right with consumers and regulators. A conventional public listing will further enhance that.
Through the Tagomi acquisition, Coinbase picked up a team who know execution technology.
Through the float, they will get balance sheet.
There are strong network effects between these business units. For example, once you have your smart-order-router and algos for institutions, you can also offer it to consumers. By having your own liquidity pool and active market-makers, you can internalise flow that you would otherwise have dispatched to other venues, and save on execution costs.
It may be a winner-takes-all market.
I am not saying they will succeed, but the sky is the limit.