To complement the excellent answer by Ofer Zeitouni, let me offer a functional analysis perspective. We want to define an integral of the following form: $\int F(W_t)dW_t=\int F(W_t)W'_tdt$, say, for a nice $F$. We can ask, generally, when is the integral $\int G(t)H(t)dt$ naturally defined? An obvious answer is: whenever $G$ belongs to some function space and $H$ belongs to the dual of that space. Then, in particular, any "reasonable" approximation scheme $\int G_n(t)H_n(t)dt$, where $G_n,H_n$ approximate $G,H$ in their corresponding spaces, will produce the same result.

Which function spaces are we talking about? Well, note that $W'_t$ only makes sense as a distribution, and $F(W_t)$ has the same regularity as $W_t$, that it, it is not smooth. Therefore, "soft" tools like Schwarz spaces will not do. A natural scale is then that of Sobolev spaces; a function $f$ is in $H^s$ if $(1+|\xi|^2)^\frac{s}{2}\hat{f}(\xi)$ is square integrable; to a very rough first approximation this means that $f$ is (almost) s-Holder continuous. It is then clear that the dual of $H^s$ is $H^{-s}$, and that differentiation takes away $1$ from $s$. This implies that the integral $\int F(W_t)W'_t dt$ would be naturally defined if we had $W_t\in H^s$ for some $s\geq\frac12$. But by Wiener's construction, we control the Fourier coefficients very well: we know that $\hat{W}(n)=\text{sgn}(n)n^{-1}\zeta_{|n|}$, where $\zeta_n$ are i. i. d. Gaussians, and so, we are essentially asking for which $s$ does the series $\sum_n \zeta^2_n (1+n^2)^sn^{-2}$ converge. The answer is provided by Kolmogorov's three series theorem: it converges (almost surely) if and only if the series of expectations and of variances converge, which happens if and only if $s<\frac12$. So, the condition we are after fails just barely.

This explains why we cannot define the integral $\int F(W_t)dW_t$ "pathwise" just by applying Riemann, Lebesgue or whatever integration to each realization. But also the fact that the required condition is just barely missed indicates why something like Ito integration, exploiting randomness additionally, has a chance to work.

thinkyou can find a discussion in Hagen Kleinert's path integral bible. I also see a discussion in M Chaichian, A Demichev, "Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics" section 2.2.5 $\endgroup$ – AlexArvanitakis Jan 8 '20 at 23:35