Why doesn't law of large numbers apply to this stationary time-series?
I believe that you are thinking that each value of $X_t$ is determined by a different realisation of $Y$, which in this example is not true.

Suppose that $Y$ is the value that comes out from a dice throw. Thus, $Y$ can achieve any integer value between $1$ and $6$. Throw the dice once. Suppose that you get $2$. Then, according to the definition in the example, your time-series would be:

$$X_t=2 \qquad {\text{for all }}t$$

What I think the quote from Wikipedia is saying is that the mean value of this stochastic process doesn't coincide with the mean value of the random variable $Y$, as

$$\mathbb{E}[X_t]=2 \neq 3.5 = \mathbb{E}[Y]$$

If you constructed the time-series letting each value of $X_t$ be determined by a new throw of the dice, then the means would match.