Question of the Day

This question comes curtesy of listening to the Feynman lectures on physics during dental work and at 5 a.m. driving people to the airport. Nonetheless, it has been vexing me for days, so I'm putting out a call for answers, lest my brain explode under the weight of the pondering. If you can help me, or know someone who can, please send them my way [jharriett {replace this with the at sign} gmail {replace with the dot} com].

Background Point #1
Experiments have shown that a clock on an aircraft moving at high speeds will run slower than a clock on the ground. This has been interpreted as proving Einstein's theory of relativity, which apparently (I've never actually read it) says that time slows down at high speeds. This nicely facilitates time travel, including the entire Buck Rogers franchise.

Background Point #2
Back in the day, maritime navigation was a dicey proposition because calculating longitude required an accurate chronometer (e.g. a clock), and pendulum-based chronometers that kept perfectly decent time on land were unreliable at sea because the motion of the boat interfered with the motion of the pendulum. Since then, we have developed chronometers that are not affected by the motions of ships at sea. Nonetheless, every method of keeping time depends on the periodic nature of some motion (e.g. pendulum swing, coiled mainspring acting on a lever, electric pulses from a quartz crystal, the oscillations of a cesium atom).

Background Point #3
Now, obviously Albert Einstein did not get into a supersonic jet, fly around for fourteen hours, and notice that his watch was twelve billionths of a second off, then came up with the theory of relativity to explain it all. He came up with the whole concept on paper, and now we can do experiments that support it. Obviously, there is something to this, but I really don't have the theoretical physics mind to work it out myself.

Pre-Question Point #4
Without getting too technical (partly because I can't put up diagrams), every explanation I've seen for the time slowing down phenomenon basically comes down to the effect of high speed on the measuring device. Photons have longer distances to travel when the thing it's oscillating around is moving really fast, that sort of thing. Is it possible that we're just proving that there is still a mechanical flaw in the way we measure time? After all, no one thought that time actually slowed on the high seas when the clocks came back off.

Which brings me to my question:
If we could develop a method of measuring time that was completely independent of motion so the chronometer would show an accurate time no matter what the forward velocity, would time still slow down at high speeds? Does time objectively slow down or do we just lack a way of measuring time that is not influenced by high speeds?

Congratulations to anyone who has made it all the way through this post. Double congrats if it makes any sense. Triple congrats and a dozen homemade cookies if you can help me out with an answer. Quadruple congrats and a fresh apple pie for those of you who have to live with me and my asking out-of-nowhere questions on theoretical physics, just as long as you don't cut off my Science Channel. Emp. Peng. puts up with a lot, doesn't he?