Confusion about the Hiring Problem
Let us denote by $x_1,\ldots,x_n$ the qualities of the $n$ persons; so the $x_i$ are iid random variables distributed uniformly over $[0,1]$.
If you know $x_1,\ldots,x_i$, then $X_i$ becomes deterministic. For example, if $x_1 = 0.3$ and $x_2 = 0.5$ then $X_2 = 1$, whereas if $x_1 = x_2 = 0.4$ and $x_3 = 0.1$ then $X_3 = 0$. If you know only some of $x_1,\ldots,x_i$, then you can calculate the distribution of $X_i$, as your example demonstrates.
None of this is needed to calculate $\mathbb{E}[X]$, however. Using linearity of expectation, we know that
$$
\mathbb{E}[X] = \sum_{i=1}^n \mathbb{E}[X_i] = \sum_{i=1}^n \Pr[X_i = 1].
$$
In other words, in order to calculate the expected number of people hired, all we need to do is to calculate the probability that $X_i = 1$. This probability is $1/i$. You're right that the probability changes if we know more about the $x_j$, but so what? We are not interested in calculating $\mathbb{E}[X]$ given partial knowledge of the $x_j$. Instead, we want to calculate the expected value of $X$ when $x_1,\ldots,x_n$ are chosen uniformly at random from $[0,1]$, independently. This is the relevant distribution for calculating $\Pr[X_i = 1]$.
What might be throwing you off is the trick of using linearity of expectation. It allows us to treat the random variables $X_i$ completely independently, although the variables themselves are certainly dependent.