The expected value of the product of two independent random variables is the product of the expected values of each; this concludes my proof that betting everything on each round is expected value maximizing in a finite game (and infinite too, if you adopt the common ways to make "infinite" precise). I'm surprised the dialogue got that far without this being brought up!
The real heart of this game is that the trades are not zero sum between the two parties. If you have 5 of a suit and your counterparty has 1, there are huge gains from trade to be had by interacting. This is a totally necessary condition for the game to be interesting.