Your second example contradicts the first. "Sending a message" doesn't happen in an instant. There's always some latency, which can be significant in email systems (the most common, universally accepted way to send a message) and SMS.
The goal in messaging is almost always not about the send time (who cares?), but the receive time, when it is available to be read. If the goal is to have the message received at 1:00, then, as you note in the second example, sending it precisely at 1:00:00.000 guarantees it will be received late.
In any case, if you're that focused on intra-minute precision, you should probably be relying on clocks that incorporate seconds anyway.
There have been times when my wife and I resorted to including send times, because the latency between our phones was ridiculously high. I really wish SMS and email would include both sent and received times, so that latency issues can be accounted for.
The goal in messaging is almost always not about the send time (who cares?), but the receive time, when it is available to be read. If the goal is to have the message received at 1:00, then, as you note in the second example, sending it precisely at 1:00:00.000 guarantees it will be received late.
In any case, if you're that focused on intra-minute precision, you should probably be relying on clocks that incorporate seconds anyway.