In Physics one constructs the path integral by a limiting process together with a discretization procedure.
Now, in order to better paralell with the Wiener measure, consider this in Euclidean signature. What Physicists find is that the Euclidean time evolution operator $K_E(q,t;q_0,t_0)$ can be written as
$$K_E(q,t;q_0,t_0)=\int e^{-S_E[x(t)]} \mathfrak{D}x(t)\tag{1}$$
where Physicists define the right hand side of (1) by a discretization procedure together with a limit. More precisely, it is something of the form
$$K_E(q,t;q_0,t_0)=\lim_{N\to \infty}C_N\int \prod_{k=1}^N dq_k e^{-S_{E,N}(q_0,\dots, q_N,\dot{q}_0,\dots, \dot{q}_N)} $$
Here $C_N$ is a normalization factor, $S_{E,N}$ is the discretized euclidean action.
At least from one heuristic point of view, the idea of Physicists seems to be something along the lines:
A path is comprised of infinitely many points. Now, pick $N$ of these, if $N$ is large this collection of points well approximate the path. If thus we pick $t_0,\dots, t_N$ values of the parameter and vary the associated $q_0,\dots, q_N$ along all possible value, we should obtain all paths, hence we should be integrating with respect to the paths there.
Although this seems to be the idea conveyed in all Physics texts I find it remarkable that someone is able to produce any mathematics whatsoever with this.
My question here is: is there some construction of the rigorous Wiener measure based on some limiting process together with discretizations, that in the end parallels the Physicist approach?
There is a not only rigorous but also physically intuitive construction of this 1d Euclidean path integral in section X.11 "The Feynman-Kac formula" of Reed and Simon Vol. II. The key property, in the OP's notation is $$ \int K_E(q_3,t_3;q_2,t_2)K_E(q_2,t_2;q_1,t_1)\ dq_2 \ = K_E(q_3,t_3;q_1,t_1) $$ which is the semigroup property for the heat equation, or a Markovian case of the Chapman-Kolmogorov Equation. With that it is easy to construct the measure on $\mathbb{R}^{[0,\infty)}$ using the Daniell-Kolmogorov Extension Theorem. The difficulty though is to restrict this measure to a space of continuous functions. This last step typically requires the dyadic grid mentioned by user10354138.