The short answer is that there is nothing that one can't do with the EM category $\mathbf{Meas}^{\mathcal{G}}$ that one can do with the Kleisi category $\mathbf{Meas}_{\mathcal{G}}$ of the Giry monad. But the purpose of Markov categories is the capture certain aspects of probability and statistics which can then be applied to other categories without directly requiring measurable spaces. The axioms for Markov categories do indeed look like they were derived with the Kleisi category in mind because (I presume) the progenitors of that theory believe those are precisely the axioms which are necessary to model the fundamental aspects of probability and statistics. Whether those axioms are the appropriate ones or not is open to debate, but the idea and intent is to abstract the basic axioms much like one does with characterizing Abelian categories for which the axioms are applicable to a wide variety of situations. In that sense the theory is successful drawing much interest (and that is how math research progresses forward).

If one had extracted the axioms based upon the properties of the EM category then the axioms would be quite different. In referring to the EM category care must be exercised because axioms characterizing the EM category are descriptive in nature and not constructive. Those axioms say nothing about the existence of algebras (which are the objects of the EM category) outside the free algebras $(\mathcal{G}(X), \mu_X)$, and the full subcategory of all those objects is precisely the Kleisi category of the $\mathcal{G}$-monad. Proving the existence/non-existence of non free $\mathcal{G}$-algebras is the key aspect in deriving the EM category, and for the general case $\mathbf{Meas}$ we cannot say too much because of pathological spaces. Thus we must temper our ambitions and address the $\mathcal{G}$-algebras for subcategories of $\mathbf{Meas}$ which have nice properties. In practice, the category of standard measurable spaces, $\mathbf{Std}$, covers most measurable spaces which arise in practice. For that case what we do know is that the $\mathcal{G}$-algebras are isomorphic to a subcategory of the category of super convex spaces. The discussion on the page Giry monad discussing the algebras gives some arguments why super convex spaces arise in the analysis of algebras. I must emphasize that that work on $\mathcal{G}$-algebras for $\mathbf{Std}$ is still a work in progress.
A list of some known algebras for other nice subcategories of $\mathbf{Meas}$ can be found at probability monads.
The reason that the category of super convex spaces, $\mathbf{SCvx}$, is of particular interest is that in attempting to find the algebras for the case $\mathbf{Std}$ there, we find that all other algebras are super convex spaces also.

To answer your question

why aren't EM categories used?

I would say that EM categories have not been used in the past because the algebras for the subcategories of $\mathbf{Meas}$ that we do understand are not practical for applications because they were based upon subcategories using the Borel $\sigma$-algebra of a topological space, e.g., the algebras for Polish Spaces with continuous maps, $\mathbf{Pol}$, or metric spaces with short maps, $\mathbf{Met}$, all require the algebras to have continuous maps. Thus, there are no discrete algebras, e.g., $\epsilon_2: \mathcal{G}(2) \rightarrow 2$ given by $\epsilon_2(p \delta_0 + (1-p) \delta_1)= 0$ for all $p \in (0,1]$ and $=1$ for $p=0$ is a $\mathcal{G}$-algebra in $\mathbf{Std}$, but it is not an algebra in $\mathbf{Pol}$ or $\mathbf{Met}$. Thus those algebras are not practical for applications. That is why the $\mathcal{G}$-algebras for $\mathbf{Std}$ which only requires the $\mathcal{G}$-algebras to be measurable is so important.

Whether the category of $\mathcal{G}$-algebras affects the synthetic approach to probability and statistics is still to be discovered. But Markov categories have attracted the lions share of research effort and the categorical understanding of probability theory has made progress based upon those axioms.