On the day DST starts, that event cannot exist, at 01:59 clocks will jump forward to 03:00. Vice versa, on the day DST ends, any time between 02:00 and 02:59 will happen twice.
You might not like DST, but leaving it in the hands of every entity to interpret is a far worse situation.
Imagine if there was a powerful body representing the population that could decree a change like this!
Instead, we only have a body powerful enough to decree that all of those people change their clocks on the same day.
Anyway, the best part of changing the hours instead of the clock is that not everybody needs to do it on the same day.
Don't scapegoat it too much: The underlying problem of time-shifts will still exist, and that edge-case still needs to be handled by other means.
DST just makes shifts more frequent and obvious. One might even say that it draws an important amount of attention to a broader class of problems.
These days, you could have a global company where some emergency occurs and you have to call a meeting at sometime on Saturday wrt your location timezone, but one of the other locations is several hours ahead of your timezone such that the meeting occurs during/around the transition period for their location's timezone.
So, not common at all. But it may happen. That's why they choose the transition times to occur when most of the LOCAL population DOES NOT schedule meetings.
I don't think there were any Saturday/Sunday virtual meetings when this timezone stuff was created... Simpler times...