Well isn't that the point? The calculations used for knowing what time the sun will set come from the same observatories and organisations that can tell you whether the moon will be visible or not. Men have been put on the moon and rockets have been catapulted around planets into outer space based on these kinds of calculations, you don't need to take the word of some guys halfway across the world because we know, right down to the minute when the sun and the moon will set and whether it will be visible.
Ignoring all of this stuff is utterly baffling.
All that doesn't matter, as far as global sighting is confirmed. In fact, even if calculations say that the moon could be sighted, the Qur'an specifically mentions visually seeing the moon. If it cannot be seen, then you simply add an additional day to the month and finish 30 days. For example, if it is cloudy in a place like the UK, where the new moon would be impossible to sight even though calculations say that it exists, it has to be noted as not seen and an additional day is added.
The Prophet PBUH said, "Whenever you sight the new moon (of the month of Ramadan) observe fast, and when you sight it (the new moon of Shawwal) break it, and if the sky is cloudy for you, then observe fast for thirty days."
The Prophet PBUH said, "Do not start the fast or break it until you see the new moon. If the new moon is obscured from you, then complete a full thirty days."
Note how the second hadeeth says that if the moon is obscured from you. So, even calculations won't actually let us see the moon if it is obscured (say on a cloudy night or for some other reason).
Another thing to note is what should we put as the cut off point geographically? For example, the moon was not sighted in Canada, but it was sighted in the Caribbean, yet all moon sighting based committees and agencies have accepted the Caribbean account and declared Eid today. If we go merely by calculations, there are some areas that it would be a difference of a few kilometres between places where you could and couldn't see the moon.
Calculations are nothing new. They are not advanced either. They have simply not been used as the primary method because they are not part of the sunnah.
And in practice? I'm having a hard time picturing guys in Al-Andalus getting news that the moon was sighted in Kenya or Somalia that same day.
Well, if we can bring up calculations and ergo technology, then we have the Internet, telephones, etc. All these newfangled ways of knowing if the moon has been reliably sighted anywhere on the planet so why not use that?
The best solution would be a global moon sighting committee that would declare the months and days based on reliable accounts, allowing for synchronized days of Eid, Ramadhan, etc. but the biggest problem is the mostly pointless divisions in the ummah.