I believe that all the exchanges want to get into more "fiat" roads. Coinbase show from the start that will move to specific areas regarding markets and after their NASDAQ entry people just follow. They are the first after all. Not to mention that Elon Musk bought "his" BTC from them and this is pretty much enough advertisement as the hype of almost all the major coins.
At least Coinbase gave ~170.000 shares to its staff/ employees...
I also believe that going public made it look more professional and more importantly more legal. Believe or not there are still people who think that bitcoin is a gray area in law, it is not illegal but it is not legal neither, it is just not in the law at all. We know the reality, it is in the law, it is legal, it is regulated, everything is perfect, a company can't get that big without doing something legal, so we all knew the real answer, but there were tons of people who didn't know if it was legal or not.
With an IPO you are basically telling the whole world, government and it is financial branches all checked from bottom to top every single data there is financial or not, and couldn't find anything illegal, so we are going to be open to public now, and saying that made a lot of people feel more comfortable using it. That is at least what I believe some parts of this was.