What's wrong with this stochastic integral calculation?

Question

To compute $E[(\int_0^1 W(s) ds)^2]$, I know that using integration by parts and Ito isometry, we have: \begin{align} E\left[\left(\int_0^1 W(s) ds\right)^2\right] &= E\left[\left(W(1) - \int_0^1 sdW(s)\right)^2\right]\\ & = E\left[\left(\int_0^1dW(s) - \int_0^1 sdW(s)\right)^2\right]\\ & = E\left[\left(\int_0^1 (1-s) dW(s)\right)^2\right]\\ & = E\left[\int_0^1 (1-s)^2 ds\right]\\ & = \frac13 \end{align} But what is wrong with the following calculation? \begin{align} E\left[\left(\int_0^1 W(s) ds\right)^2\right] &= E\left[\left(W(1) - \int_0^1 sdW(s)\right)^2\right]\\ & = E\left[W(1)^2 - 2W(1)\int_0^1 s dW(s) + \left(\int_0^1 sdW(s)\right)^2\right]\\ &= 1 + E\left[\int_0^1 s^2 ds\right]\\ & = \frac43 \end{align}