Why don't we allow the canonical Gaussian distribution in infinite dimensional Hilbert space?

Here is a more "probabilistic" argument why no such distribution can exist.

Let $H$ be an infinite dimensional Hilbert space equipped with a Gaussian Borel measure $\mu$. Suppose the covariance matrix of $\mu$ is the identity. Let $\{e_1, e_2, \dots\}$ be an orthonormal sequence. Then if we define $X_i : H \to \mathbb{R}$ as the continuous linear functional $X_i(x) = \langle x, e_i \rangle$, $X_i$ is a Gaussian random variable on the probability space $(H, \mu)$ with mean 0 and variance 1. Moreover, since the $X_i$ are jointly Gaussian distributed and are uncorrelated (because the $e_i$ were orthogonal), they are independent. So $\{X_i\}$ is an iid $N(0,1)$ sequence.

However, for any $x \in H$, we have $$\sum_i |X_i(x)|^2 = \sum_i |\langle x, e_i \rangle|^2 \le ||x||^2 < \infty$$ by Bessel's inequality. In particular, $X_i \to 0$ surely. This is absurd for an iid sequence (by any of several possible arguments).

Such a thing is studied. But (of course) it is not a measure on the Hilbert space: It assigns measure zero to every bounded set; but the whole space, although a countable union of bounded sets, has measure one... This type of thing is studied under the name "cylindrical measure". You can find many books on the subject.