So that dates astrology's interest in the time of birth to at the latest 350 years ago.

As one previously invited not to comment in this thread, I hesitate to post again ... but I didn't hesitate very long!

Although accurate clocks only go back 350 years or so, it doesn't follow that astrology can only have regarded time of birth as significant for that long. It would be possible to make the case that the difference is one of degree: You can get one quality of analysis based on knowing only the day, a different quality knowing the time as roughly early morning, late morning, afternoon, evening or night, another quality again when the precise hour is known, yet another when we know the time down to the second and, who knows, even the millisecond.

As you know, I don't subscribe to astrology at all, but they is no inconsistancy between requiring an accurate time nowadays with the limitation on knowing the time accurately way back when.