Afternoon,
Purely based on my own experience, as a non-engineer, I would suggest that drilling on the lathe is going to be more accurate than using a drill press. You've already pointed out the 60mm drop being problematic, which doesn't exist when using lathe. Also you're more likely to have an accurate and heavy wood chuck to hold the wood when compared to the free standing chucks used for presses.
But in saying all this, which ever works for you is the way to go.
Chriscb