Maybe new words would fix it.

But even from the historical perspective it doesn't make sense. Why wouldn't they pick dawn as the natural starting point of the photoperiod? As you said, nobody was awake at night, so why did they choose a time when nobody was awake to make the differentiation on the date?
When you say "sunrise varies quite a bit", that's only from the perspective of a midnight-centric time measurement; sunrise wouldn't vary, it'd be the start of the day by definition.

There are some issues with using dawn, but they wouldn't be a concern historically and we have modern solutions;
Like days wouldn't be exactly 24 hours, and dawn is affected not just by latitude but also geography.

But fundamentally it's more satisfying if a calendar day is compromised of one single contiguous day and one contiguous night 😌