The Yamabe flow is (up to a constant) the gradient flow of the Yamabe functional on the unit volume conformal class, as you expected. The comment by @Mark Peletier hints at your error: you aren't using the correct "inner product."

We briefly discuss the Ebin metric on the space of all metrics $Met$. Recall that $T_gMet = Sym^2 T^*M$. Then the Ebin/$L^2$ metric at $g$ is defined by
$$
g_E(h_1,h_2)|_g = \int_M tr(g^{-1}h_1g^{-1}h_2).
$$
See e.g. http://arxiv.org/pdf/0904.0177v1.pdf for more information on the $L^2$ metric.
Here, we are interested in the restriction of the Ebin/$L^2$ metric to the unit volume conformal class $[g]_1$. Now, the correct statement is

The Yamabe flow is (up to a constant multiple) the (negative) Ebin/$L^2$-gradient flow of the Yamabe functional on $[g]_1$.

First, note that the tangent space to $[g]_1$ at $g$ is
$$
T_g[g]_1 = \left\{w g : \int_M w dV_g = 0\right\}.
$$
and the Ebin, or $L^2$ metric restricted to $[g]_1$ is given by
\begin{align*}
g_E(w_1g,w_2g)|_g & = \int_M tr(g^{-1} w_1 g g^{-1} w_2g)\\
& = \sum_{i,j=1}^n \int_M w_1 w_2 g^{ij}g_{j}^{k} g_{kl}g^l_i \, dV_g\\
& = n \int_M w_1 w_2 dV_g.
\end{align*}
So, up to a constant (which we'll ignore), $g_E|_{[g]_1}$ at $g$ is the $L^2$ inner product of the conformal factor.
\begin{align*}
\frac{d}{dt}\Big|_{t=0} Y((1+tw)^{N-2}g) & = c \int_M (R_{g}-r_{g})wdV_g\\
& = g_E((R_g-r_g)g,wg)|_g
\end{align*}
This shows that (up to a constant),
$$
\nabla_{[g]_1} Y|_g = c_n(R_g-r_g)g,
$$
which is the Yamabe flow.

If you would rather think of the flow as a flow on the level of conformal factors, you may be a bit dissatisfied with the previous computation. So, lets do it again, where we imagine that $g$ is fixed, and the Yamabe flow at time $t$ is given by $v^{N-2}g$ (recall that $N = \frac{2n}{n-2}$). Then,
$$
T_{v^{N-2}g}[g]_1 = \left\{w v^{N-3} g : \int_M w v^{N-1} dV_g = 0\right\},
$$
so
\begin{align*}
g_E(w_1 v^{N-3} g,w_2 v^{N-3} g)|_{v^{N-2}g} & = \int_M tr( v^{2-N} g^{-1} w_1 v^{N-3} g v^{2-N} g^{-1}w_2 v^{N-3} g ) v^N dV_g\\
& = n \int_M w_1 w_2 v^{4-2N+2N-6+N} dV_g\\
& = n \int w_1 w_2 v^{N-2} dV_g.
\end{align*}
Moreover,
\begin{align*}
\frac{d}{dt}\Big|_{t=0} Y((v+tw)^{N-2}g) & = c\int_M (R_{v^{N-2}g} -r_{v^{N-2}g})v^{N-1}w dV_g\\
& = c\int_M (R_{v^{N-2}g} -r_{v^{N-2}g})v w v^{N-2} dV_g\\
& =c g_E((R_{v^{N-2}g} -r_{v^{N-2}g})v^{N-2}g,w v^{N-3}g)|_{v^{N-2}g}
\end{align*}
So,
$$
\nabla_{[g]_1} Y|_{v^{N-2}g} = (R_{v^{N-2}g} -r_{v^{N-2}g})v^{N-2}g,
$$
which of course is what we expected.