...Being equal to the speed of a starship that is departing a busy starbase might not mean being equal to the top speed of a starship, or even to her cruising speed.
Generally, the TOS starship departed places at warp 1 or 2. She was seldom heading away from the week's adventure at a high warp factor in the end of an episode, even though it would stand to reason that she would soon be utilizing warp 6 in order to get to the next adventure. A slow start would make sense as a standard procedure - and would give Kirk some slim hope of catching Spock by flying a shuttle that can do at most warp 2.
Well, most backstage sources suggest the Type-7 shuttlecraft is equipped with a warp core to power those nacelles, so if it isn't warp capable it must be pretty damn fast.
True. But it need not be at warp speed in episodes like "Samaritan Snare" where the windows show impulse views.
We already have a reason to believe that warping is sometimes inadvisable or flat out impossible when approaching a planet. "Best of Both Worlds" is a perfect example: the world is about to end, but Riker slows to impulse in his quest to reach Earth before the Borg. If we consider this to be a common occurrence, it would make sense for a starship not to go all the way into a "no-warp" star system, but rather to drop off the passengers at the edge of the system and let them continue the voyage aboard a shuttle. And never mind that the shuttle would be capable of warp - it would be flying at impulse because of the "no-warp" condition, and would be a more affordable vehicle for wasting time in that fashion than an entire starship.