The first time I taught myself rigorous measure theory, I used the "linear functionals on $C(X)$" approach for compact Hausdorff spaces, so I got first-hand knowledge of where it doesn't work for probability.

Consider the simplest case of the strong law of large numbers:

The sample average of a sequence of independent fair coin flips converges almost surely to $\frac{1}{2}$.

Which is to say, the following set has measure $1$ under the independent product measure of Bernoulli trials with equal probability for $0$ and $1$:
$$
M = \left\{ (x_n)_n \in 2^{\mathbb{N}} \,\left\lvert\, \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{2} \right. \right\}
$$
(since this sometimes causes confusion, consider $2 = \{0,1\} \subseteq \mathbb{R}$ to interpret the above sum)

Using the linear functionals on $C(X)$ approach, one can only take the measure of a set $S$ whose indicator function $\chi_S$ is continuous, *i.e.* a clopen set. A clopen set in $2^{\mathbb{N}}$ can only restrict the value of a sequence $(x_n)_{n \in \mathbb{N}}$ at finitely many coordinates $i_1, \ldots, i_k \in \mathbb{N}$. One can therefore define a sequence $(y_n)_n$ satisfying these conditions at $i_1,\ldots,i_k$ and alternating back and forth between $0$ and $1$ for all $i > i_k$, making the sample average converge to $\frac{1}{2}$, and thereby showing that $M$ nontrivially intersects every clopen set, and is therefore a dense set with empty interior, so we can't express its measure. So the strong law of large numbers and related theorems such as the pointwise ergodic theorem are not doable. This also shows the inadequacy of measure theory based on valuations on open sets or lower semicontinuous functions for this application.

Here are some other points against the Bourbaki approach:

Locally compact Hausdorff spaces are not closed under infinite products. In fact, a product of locally compact Hausdorff spaces is locally compact iff all but finitely many of the factors is compact. So the spaces of sequences $\mathbb{N}^{\mathbb{N}}$ and $\mathbb{R}^{\mathbb{N}}$ are not locally compact. However, for probability theory this can be resolved using linear functionals on $C_b(X)$ (the space of bounded continuous functions) equipped with a certain topology (the 'strict topology', the finest locally convex topology $\mathcal{T}$ that agrees with the topology of uniform convergence on compact subsets of $X$ when restricted to norm-bounded subsets of $C_b(X)$). This also takes care of Wiener measure, white noise, and the measures on distributions used in bosonic quantum field theory (and moreover, 'measures' in fermionic quantum field theory must be viewed as functionals (on anticommuting algebras), because we don't have 'fermionic sets').

All uncountable Polish spaces are measurably isomorphic to each other and to $[0,1]$, $\mathbb{R}$, $\mathbb{R}^2$, $2^{\mathbb{N}}$, $\mathbb{N}^{\mathbb{N}}$, $\mathbb{R}^{\mathbb{N}}$. If you have a purely topological measure theory, you can't take advantage of this fact to reduce a measure-theoretic question on one space to a question that's easier to solve on one of these spaces using the other structures they have available.

I still advocate using the linear functional approach it when it works. I think at the time Bourbaki wrote the book, a lot of people were unduly resistant to using linear functionals to define measures, and it affords a lot of simplifications under certain circumstances (distribution theory, Grothendieck's work on inequalities and integral operators, C*-algebras, Haar measure - not coincidentally, areas important to Dieudonné, Schwartz and Weil, and the application to probability monads was important for my own work).

Another useful, if neglected, point of view is finitely-additive measures on the clopen sets of a Stone space, as these faithfully represent Baire measures and Radon Borel measures on that space just as well as linear functionals on the continuous functions. The infinite probability spaces one first encounters, *e.g.* $2^{\mathbb{N}}$ are Stone spaces. But this isn't good enough if you need *e.g.* the central limit theorem, which depends on the topology of $\mathbb{R}$.

I advocate against fixing a single definition as *the* definition, to be used at all times, forsaking all others, because different applications suggest the use of different definitions, overlapping but not quite always equivalent, for different purposes.

measures as functionalsapproach would not be insufficient generality but rather excessive and unnecessary abstraction (an unease not alleviated by your reference to Bourbaki). $\endgroup$