Once a payload is integrated and the LV has stood up on the pad, is a solid-fueled rocket actually any faster to orbit than a liquid one?
In flight time, no. The time it takes to get from liftoff to orbit depends on your acceleration, which depends on the power-to-weight ratio of the rocket.
A high acceleration is usually avoided for space launch vehicles, because it provides no advantage and costs more. A high acceleration requires a stronger structure for both payload and rocket, and can require a heat shield on the nose of the rocket.
There are examples of military missiles with high acceleration (the Sprint ABM accelerated at 100g). Space launchers are usually in the 1-5G bracket (depending on where in the mission profile you are).
Even a rocket like Vega that is not intended for manned use, and which accelerates quickly right on liftoff limits G-loading to about 5G.
As Antony X said, you do get an advantage in that you don't have to fuel the vehicle on the pad. The associated disadvantage is that you have to handle the solid rocket as live ammunition once it's been fueled, so it can be more cumbersome and expensive to move around.
For ICBMs, not having to fuel them immediately before launch is a big advantage, as it reduces the launch delay from hours to seconds. For space launchers, a launch campaign can take weeks or months, from rocket stages arriving at the launch site to liftoff. Most of this is spent assembling and checking the launcher.