The non-endpoint Strichartz estimates for the (linear) Schrödinger equation: $$ \|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)} $$ $$ 2 \leq q,r \leq \infty,\;\frac{2}{q}+\frac{d}{r} = \frac{d}{2},\; (q,r,d) \neq (2,\infty,2),\; q\neq 2 $$ are easily obtained using (mainly) the Hardy-Littlewood-Sobolev inequality, the endpoint case $q = 2$ is however much harder (see Keel-Tao for example.)

Playing around with the Fourier transform one sees that estimates for the restriction operator sometimes give estimates similar to Strichartz's. For example, the Tomas-Stein restriction theorem for the paraboloid gives: $$ \|e^{i t \Delta/2} u_0\|_{L^{2(d+2)/d}_t L^{2(d+2)/d}_x} \lesssim \|u_0\|_{L^2_x}, $$ which, interpolating with the easy bound $$ \|e^{i t \Delta/2} u_0\|_{L^{\infty}_t L^{2}_x} \lesssim \|u_0\|_{L^2_x}, $$ gives precisely Strichartz's inequality but restricted to the range $$ 2 \leq r \leq 2\frac{d+2}{d} \leq q \leq \infty. $$

As far as I know, the Tomas-Stein theorem (for the whole paraboloid) gives the restriction estimate $R_S^*(q'\to p')$ for $q' = \bigl(\frac{dp'}{d+2}\bigr)'$ (this $q$ is different from the one above), so I'm guessing that this cannot be strengthened (?).

So my question is: what's the intuition of what goes wrong when trying to prove Strichartz's estimates all the way down to the endpoints using only Fourier restriction theory?