Kolmogorov vs Ionescu-Tulcea extension theorem (again) Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there.

I've recently been trying to understand the difference, both theoretical and practical, between the KET and Ionescu-Tulcea extension theorem (ITET for short).
Most of the answers I found stress the fact, that ITET does not impose topological restrictions on the state space. In the book Almost None of the Theory of Stochastic Processes the authors write that some probabilists think that topological conditions are not natural in the probabilistic world:

Borel spaces are good enough for most situations we find ourselves modelling, so the Daniell-Kolmogorov Extension Theorem (as it's often known) see a lot of work. Still, some people dislike having to make topological assumptions to solve probabilistic problems; it seems inelegant. The Ionescu-Tulcea Extension Theorem provides a purely probabilistic solution, available if we can write down the FDDs recursively, in terms of regular conditional probability distributions, even if the spaces where the process has its coordinates are not nice and Borel.

This leads to my first question. Is it really a matter of non-naturalness of topological assumptions on measurable spaces? I see how this argument makes sense, but I've never seen a convincing example where some topology cannot be arranged.
I also know that there are several quasi-topological sufficient spaces' properties which make KET work. Most common reference is to the notion of prefect spaces, but in the same book mentioned above the authors say there is a necessary and sufficient condition:

Notice that the Daniell, Kolmogorov and Ionescu-Tulcea Extension Theorems all give sufficient conditions for the existence of stochastic processes, not necessary ones. The necessary and sufficient condition for extending the FDDs to a process probability measure is something called $\sigma$-smoothness (see Pollard (2002) for details).

I did not have time to read about this $\sigma$-smoothness condition, but I guess it's also something quasi-topological and similar to perfection.

There is, however, a much more straightforward difference between the two. Klenke in Probability Theory. A Comprehensive Course describes the difference between KET and ITET as the difference between discrete and continuous cases:

In this section, we first show how to implement countably many successive experiments on one probability space (Ionescu-Tulcea's theorem). Thereafter, we also construct probability measures on products of uncountably many spaces (Kolmogorov's extension theorem).

The second question. Would it be right to say that this is the main difference? Is there an uncountable extension of ITET? I couldn't find one... It also doesn't seem possible to chain kernels in the spirit of ITET even when the process is indexed by $\mathbb{Q}$. It seems to me that any extension of ITET would at least require some natural well-order on the index set (in the sense that for $\tau \in T$ there is a well-defined unique successor).
The third question. It would be natural to think that the necessity of topological assumptions in KET and the absence thereof in ITET is due to the fact that ITET only works with processes indexed by $\mathbb{N}$, while KET does not impose assumptions on the index set. Is it true that the difficulties making topological assumptions of KET inevitable arise from here?

The third big difference between KET and ITET is that they start off from different objects. KET works with fdds, ITET with chains of kernels. Since we can always build fdds by chaining kernels, but not the other way around, an analogy pops to mind. The Carathéodory extension theorem starts with either an abstract outer measure or an induced by a pre-measure one. The induced outer measures, as far as I understand, possess a few nice properties by birth, which makes proofs easier for them and also allows to dispense with some assumptions.
My fourth question therefore is this: is this analogy valid? Can the difference between the two results be attributed to the fact that abstract fdds are more complicated than the ones induced by a chain of kernels?
 A: 1. In a sense, at least in discrete time, ITET is more general than KET. Under typical hypotheses for KET, a disintegration theorem can be applied to obtain, from the consistent family of probability measures that form the data for KET, an associated family of conditional distribution to which ITET can be applied.
2. Even the countability of the index set in ITET is somewhat illusory. For example, let $(E,\mathcal E)$ be a measurable space, let $T$ be an index set, and let $\{\mu_F: F\subset T,|F|<\infty\}$ be a consistent family of probability measures indexed by the finite subsets of $T$, each $\mu_F$ a probability measure on the product of $|F|$ copies of $(E,\mathcal E)$. Let $P$ be the Kolmogorov measure on the $T$-fold product $(\Omega,\mathcal F)$ of the $(E,\mathcal E)$, with coordinate maps $X_t(\omega)=\omega(t)$ for $\omega\in\Omega$ and $t\in T$. Consider now an event $A\in\mathcal F=\sigma\{X_t:t\in T\}$. Then there is a countable set $J\subset T$ such that $A\in\sigma\{X_t:t\in J\}$. The value of $P(A)$ is therefore determined by the subfamily $\{\mu_F: F\subset J,|F|<\infty\}$, by application of KET or ITET, as may be appropriate. Of course, to apply ITET you would need to order $J$ and then to produce the associated collection of conditional distributions.
A: I don't see how topological (or other, more rigid, structural) assumptions can be viewed as inelegant.
To the contrary, studies of probability distributions only on spaces with no or little structure will clearly be not as rich.
There has to be some sweet spot where the theory is rich and interesting enough, but the setting is not too specific.
As for the ITET, excluding continuous-time random processes does not look very appealing.
