Actually, I think the real reason why the shuttle has a non vertical start was not cited here.
If you fly vertically, for each layer i of the atmosphere, you cross a certain thickness Ti of atmosphere.
If you fly with an angle to vertical theta > 0, then you cross a thickness Ti / cos(theta)
Since the cosine function is flat around 0, ( no first order term in the taylor series, 1-theta²/2 ), then having a small angle theta does not change a lot the crossed thickness. It's even almost free for the first degrees. For instance, with theta=5°, you cross a thickness of Ti + 0,38%
However, the horizontal speed gained is in sin(theta), linearly growing with theta for small angles. This means that the first degrees give a significant advantage as you are already gaining horizontal velocity. For 5°, you give 8.7% of your acceleration to horizontal velocity. That's a lot.
This means, that the optimal start angle is greater than zero. The actual one is complex to find, but apparently, for the shuttle, it was something like 20°.
Why isn't this used for rockets? Here I'm less sure, but maybe that oblique flight at rather low velocity is not easy to stabilize for a pure rocket with nearly no lift and control surface. For the space shuttle, it's easy.