This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that the math is wrong: it is correct. Here is a mathematical insight from Ross' work. Abstracting from the finance and economics, the purely probabilistic content of Ross' Recovery theorem is the following. Suppose we are given a positive measure that describes the product of transition frequencies
of a univariate Markov process and a positive multiplicative functional $m_t = \exp(-\int_0^t g_u du)$, where $g_u$ is some real-valued stochastic process at time $u$.
The Ross recovery theorem answers two questions posed together:
Question 1) Under what additional probabilistic structure, can we remove the multiplicative functional and thereby obtain transition frequencies, which need not be the same as the transition frequencies multiplying the functional?
Question 2) What additional probabilistic structure to Question 1 is needed to uniquely obtain these possibly different transition frequencies?
Ross' answer to both questions posed simultaneously is that the sufficient conditions are that the univariate Markov process is an irreducible
time-homogeneous finite-state Markov Chain and that the measure change be transition independent. From a purely probabilistic perspective, Hansen Scheinkman 2009 answer question 1. Not surprisingly, they can do so with weaker sufficient conditions, requiring neither finite states or transition independence. Both papers supply mathematical insights into how to change from a positive measure eg Arrow Debreu security prices to a probability measure eg forward measure or market beliefs or a third probability measure. From a purely probabilistic standpoint, Ross incremental contribution is to answer question 2. To get the stronger conclusion of uniqueness, he makes stronger assumptions. I don't personally consider this a flaw. In fact, identifying the additional structure that leads to the stronger conclusion is usually considered by mathematicians to be a mathematical insight.
Some economists argue in contrast that this additional structure is a weakness because
1) it is less general, and rules out some well known successful models, and
2) the particular application Ross suggested ends up being empirically refuted.
So to summarize, I believe that the two incremental assumptions Ross made in answering Question 2 can simultaneously be interpreted as both a mathematical insight and an economics flaw.