Grey Ghost is correct that solid propellant rocket motors cannot, in general, be throttled. (The exception to this is that both Thiokol and Aerojet worked on throttleable and restartable solid motors in the 'Sixties, which basically had a plug inside the chamber that could choke the nozzle throat and stop combustion, but it added mass and complexity with no real purpose and has never been implemented on a production motor.) For most applications for which solid motors are used–tactical rockets, small sounding rockets, and intercontinental ballistic missiles–this isn’t really a problem, although it can require the need for trajectory shaping maneuvers to expend additional energy. For something like an interceptor or space launch vehicle that has to hit a very precise pierce point or orbital apse, however, getting rid of extra energy can be crucial. For a system like the THAAD (which is the vehicle that was pictured), it is necessary to perform an energy management maneuver in order to allow the booster to deliver the interceptor to both long range and also engage targets at shorter range.
There are a few commercial motors that can be machined after casting to achieve specific thrust profiles such as the Castor 120, but most motor propellant grains are cast into the case with a mandrel and flares that form the fin, slot, and cylinder profiles, and this tooling cannot be easily modified or reconfigured. It is very elaborate, as it generally has to be removed in pieces like a jigsaw puzzle in reverse and also not damage or ignite the propellant by friction or pressure during removal. You also cannot just scrape out propellant from just any part of the grain. The burn rate is heavily dependent both on pressure and exposed surface area, and this determines the thrust profile. Removing propellant changes the total impulse more or less directly, but can also affect pressure profiles, burnback geometry, and can cause instability in combustion flows, which can cause the motor to develop thrust or vibration that exceeds what it can handle structurally, or create pressure transients that can damage the nozzle.
Most space launch vehicles are liquid propellent core vehicles that use solid rockets as strap-on boosters if at all. There are some upper stage/transtage use of solid propellant motors such as the Inertial Upper Stage that are primarily used to reduce fueling complexity (developed originally because it was decided that the Shuttle could not safely fly with a liquid propellant Centaur in its cargo bay, which would have be be defueled prior to a Return To Launch Site abort). There are a few purely solid propellant motor space launch vehicles, like the Orbital Sciences Pegasus and Taurus, the Lockheed Martin Athena, and the Peacekeeper- and Minuteman-based Minotaur family, but in general their precision is determined by evaluating motor-specific propellant load and trajectory shaping so that they do not need to be throttled.
The propellants–fuel and oxidizer–are not terribly expensive compared to the rest of the vehicle and the payload, and there would be little savings in underloading them. Trimming out large amounts of propellant, as noted, would be a hazardous operation and would require automated equipment (as is used on the Castor 120 grain machining). For tactical systems you have to have sufficient propellant load to reach your maximum range and then waste off anything that you don’t need. The largest cost in a space launch system isn’t the propellants, the motor or engine, or even the whole vehicle itself, but rather all of the support, facilities, analysis, inspection, reporting, and so forth that goes into supporting the launch, which is why efforts to make vehicles reusable without minimizing the amount of effort in supporting and refurbishing usually fail to achieve cost targets.
Stranger