Think of gravity as: if you don't move per se, you'll get pulled in...so you have to "run away" at least as fast as you're being pulled. Your "run away" speed counts as speed which, as you noted, certainly affects time.
Lewis Carroll was more correct than you thought when you read _Through_The_Looking_Glass_:
"Well, in our country," said Alice, still panting a little, "you'd generally get to somewhere else—if you run very fast for a long time, as we've been doing." "A slow sort of country!" said the Queen. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!"If we build four of these ultrasenstive clocks, spaced well apart, we could use them as a "gravitational telescope" detecting & mapping moving/changing masses by how each clock gets out of sync with the others. The real interesting question is how _small_ a gravitational shift could be detected.
That actually is fascinating. Thank you. Well stated.
Gravitation redshift is about 10^-16 per meter, so a clock accurate to one part in 10^18 would be able to detect the gravitational redshift of about 1/100 m, or 1 cm.