I found an arguably simpler answer (statement 4.3 in this paper):
http://arxiv.org/abs/1009.4079;
then again, it's arguable it just moves the difficulty elsewhere (to other results of Borel).
The trick is that the fiber restriction homomorphism is surjective if and only if the Serre spectral sequence $M \to M_T \to BT$ collapses at $E_2$, and then and only then does one have $\dim H^*(M) = \mathrm{rank}_{H^*(BT)} (H^*_T(M))$.
Since $G^T = T$, one gets $H^*_T(T) = H^*(BT) \otimes H^*(T)$, and by the Borel localization theorem applied to conjugation of $G$ by $T$, the rank over $H^*(BT)$ of $H^*_T(G)$ is also $\dim H^*(T)$; but $\dim H^*(T) = \dim H^*(G)$.