Based on the contents of this book, no—they only argue that they get the same results measure theory does. They argue that it is a better choice as a theoretical foundation on the grounds of simplicity, intuitive appeal, and occasionally having broader implications than the measure theory version. This point is addressed frequently in the reviews listed on the website, but in general if you already know measure theory there is not much advantage to learning this, except to refine the occasional subtle philosophical point. However, the book was published in 2001 and they have had continuous research since then; there may be new results of which I am unaware.
That being said, the direct deprivation of finance theory does offer some specific advantages. I don’t know how finance theory was actually developed, but my impression is it comes from probability theory—so the process changes from this:
Measure → Probability → Finance
Games → Finance
The pitch is that the probabilistic elements are in fact native to finance theory under this construction. Their central example in the book is that there are things in finance we assume to be stochastic but are not, and they argue people accept this assumption because they believe measure theory justifies it. They claim they can accommodate a mixture of stochastic and deterministic behavior without changing any assumptions. As far as I can tell this won’t enable us to calculate things we could not before, but it does seem very much in the vein of knowing where your assumptions lie, and making the model match reality as closely as possible—which hypothetically would lead to making fewer mistakes about how to calculate things.
I really should have signposted this up front, but when they are talking about game theory they really mean more in the John Conway sense them the economics sense. They actually rewind as far as the 1890s and then pick up the thread from there. Since they don’t provide historical context for mathematical or economic game theory, I think it would be fair to say that they have re-derived game theory from base assumptions. They seem to view games and probability as conceptually inseparable, because Probability arose from betting strategies in games in the first place. They argue that the prices are the probabilities; this is very similar to the way that futarchy works via the efficient market hypothesis.
I will make an update to clarify some of this!