Randomized linear search -- Why does reducing the sample size make the running time constant?
You made a simple algebraic error.  You said $\Pr[Y=i] = 1/n$, then you wrote

$$E[Y] = \sum_{i=1}^n i \Pr[Y=i] = \sum_{i=1}^n {1 \over n},$$

but that is incorrect: you omitted the $i$ in the second sum, i.e., you didn't plug in the value of $\Pr[Y=i]$ correctly.  Assuming that both indexing into the list and removing an item from the list can be done in one time step, the correct derivation is

$$E[Y] = \sum_{i=1}^n i \Pr[Y=i] = \sum_{i=1}^n i \cdot {1 \over n} = n(n+1)/2 \cdot {1 \over n} = (n+1)/2 = \Theta(n).$$