Or were all the old second counting systems wrong?
Many people have pointed out that if you start a timer on your phone and count out to 45 Mississippis, hippopotamus or number-one thousands it now ties out to a minute.
I have tried it a dozen times myself and after 45 counts I get anywhere between 54 to a minute and 4 seconds.
I specifically remember counting chunks of time as long as 15 minutes and not being off by a minute.
The limit of speed of light is a property of space-time, not a property of light.
Another way to understand it is that it is the maximum speed of causality. The limit doesn’t go only for light, but for instance also for gravity, which AFAIK is also proven by the measurement of gravitational waves.
The reason only light can achieve this speed is that it is massless. Because if light had mass, it would have infinite energy, and take infinite energy to achieve the speed of light.
There are numerous explanations for how 2 objects moving at the speed of light still only approach each other at the speed of light.
Take a look at Youtube, there are many good videos explaining relativity in general and speed of light in particular.