You don't need to invoke the speed of light anyway. If you have two extremely accurate clocks, which tick at exactly the same rate, and they are synchronised to show exactly the same time, and you keep one here on earth, and send the other up to the International Space Station and keep it there for 6 months, and then bring it back and compare the two, it will be behind the one that stayed behind, by about 0.007 seconds. It is only the fact that the effect is so small in everyday experience that stops us being aware of it. If I had two atomic clocks and sent one by train to Madrid and back, I could compare the readings and see the difference.
There is an equation for calculating the slowdown of time in a system which is moving with another system:
This says that the time passed in the moving system, "t prime" (the letter t with a little tick) is equal to:
the time passed in the stationary system (t without a prime), divided by the square root of (one minus the velocity squared over the speed of light squared)
This has been confirmed by experiments, modern ones including the Hafele-Keating experiment in 1971. Atomic clocks were flown aboard commercial airliners twice around the world, first eastward, then westward, and then compared to others that remained at the United States Naval Observatory. When reunited, the differences were consistent with the predictions of special and general relativity. In fact the effect was observed experimentally in the 1930s.
I think you are of the opinion that the non-existence of a "universal now" is against intuition, and while I can see where you are coming from, I don't think that Hafele and Keating's clocks were faulty, any more than Michelson's measurements were.