What do you mean by "so we can price as usual"? What you showed is that for every $c \in \mathbb R$ we can find a probability measure such that the drift of $S$ is $c$. But that does not really say anything about pricing.
You can easily see that $V_t = E^Q_t[e^{-r(T-t)} \Phi_T]$ does not give arbitrage free prices with your choice of $Q$. Indeed if $\Phi_T = S_T$, then $E^Q_t[e^{-r(T-t)} S_T] = S_0$. But if I invest $S_0$ in the stock at time $0$, then at time $T$, I will own $S_T$ plus all the dividends I earned.
What matters in arbitrage free pricing is the notion of self-financing portfolio (= strategy). Two self-financing portfolio giving the same cashflows must have equal price. The problem is that, in the presence of dividends, a portfolio $V_t = S_t$ is not self-financing.
Edit: You may be confused by the fact that a risk neutral measure is often defined as a measure under which the discounted underlyings price processes are martingales. If that was the case your measure would be a risk neutral measure. But the right definition of a risk neutral measure is a measure under which the self-financing portfolios discounted price processes are martingales (this is mentionned by Shreve p. 234). This is important because being self-financing is independent of the measure you consider so you can reason under the historical measure to decide whether a portfolio is self-financing or not and then use risk neutral pricing to actually price it.
This is where the pricing formula comes from: if
- the payoff $\Phi_T$ can be replicated by a self-financing portfolio $V_T = \Phi_T$,
- there exists a measure under which self-financing portfolios discounted prices are martingales
then
$$
V_t= E_t^Q[e^{-r(T-t)}V_T]= E_t^Q[e^{-r(T-t)}\Phi_T].
$$
which gives us the price of the cashflow.
Consider a market with underlying assets $(S^1,\ldots,S^d)$ and money account $S^0_t = e^{rt}$.
Claim: In the abscence of dividend, if all the discounted $S^i$ are martingales under a probability measure $\mathbb{Q}$ then so are all the discounted self-financing portfolios (so $\mathbb{Q}$ is a risk neutral measure).
Proof: Consider a portfolio
$$
V_t = \delta^0_t S^0_t + \sum_{i=1}^d \delta_t^i S^i_t
$$
Notice that $\delta^0_t S^0_t = V_t - \sum_{i=1}^d \delta_t^i S^i_t$. Plug this in the self-financing equation
\begin{eqnarray*}
dV_t &=& \delta^0_t dS^0_t + \sum_{i=1}^d \delta_t^i dS^i_t \\
dV_t &=& \delta^0_t rS^0_tdt + \sum_{i=1}^d \delta_t^i dS^i_t \\
dV_t &=& r(V_t - \sum_{i=1}^d \delta_t^i S^i_t)dt + \sum_{i=1}^d \delta_t^i dS^i_t \\
dV_t - rV_t &=& \sum_{i=1}^d \delta_t^i (dS^i_t - rS^i_t dt) \\
d(e^{-rt}V_t) &=& \sum_{i=1}^d \delta_t^i d(e^{-rt}S^i_t)
\end{eqnarray*}
Since all the $(e^{-rt}S^i_t)$ are $\mathbb{Q}$-martingales, so is $(e^{-rt}V_t)$.
Conclusion What makes risk neutral measure useful for pricing is that discounted self-financing portfolios are martingales. In the abscence of dividends, this is equivalent to the discounted underlyings being martingales. But a dividend paying asset by itself is not a self financed portfolio so it is not useful to find a probability measure under which it is a martingale.