Eid-ul-Fitr Moonsighting and Astronomical Calculations

The word 'Ro'ya' as mentioned in ahadith is not limited, in it's meaning, just to visibility and it also refers to 'knowing'. It has been used in the Qur'an many times for this meaning such as Surah al-Fil v. 1 "Have you not seen how your Lord dealt with those who had elephants?"
This refers to an event which took place before the birth of Holy Prophet (peace & blessings be upon him), so this was in knowledge and not visualised.

From the beginning and until recent times, Muslims have used visuals to determine the new moon as there was no other means to certify it. Muslims developed telescopes to help make determinations. This also shows Islam did not oppose this method. In today's day and age, there is no need for visualisation of the moon because of scientific advancement we are able to calculate with certainty the number of hours, minutes and seconds, for the new moon to be born. We have certain knowledge. Whereas there can be errors with regards to witnesses who claim to have seen the moon such as in Nigeria in 2017 which became controversial and unacceptable by groups of Imams.

Although the classical hanafi position requires actual moon-sighting, there is a Shafi' position to accept calculated methods. It is, therefore, a valid shari' opinion. Furthermore, there is no bar upon conducting ijtihad, which is inherently required due to the change of circumstances. It is the same ijtihad which has been unanimously accepted in Sahoor and Iftar calendar times that have been drawn up through astronomical calculation.

If the Muslims agree upon scientific methods then the ummah will be united on the day of Eid al-Fitr every time as it will be known well in advance. 

Ibn Taymiyyah's opinion on the Khawarij | Misguided or Disbelievers?

AUTHOR'S NOTE
"Anybody who develops an interpretation such a considering the killing of innocent people to be a noble act which will take him to paradise, or doing takfir on Muslims; is an act of disbelief. There is no Islamic funeral (Janaza) for disbelievers." 

Ibn Taymiyyah makes the point that in this particular section he admits there are two opinions on the matter and that the first is that "they are disbelievers like apostates". (Majmu'a al-Fatawa 28:518)

It is incorrect for people to make the assumption that due to Ibn Taymiyyah quoting a historic event in his works therefore he was also of this view. The event is as follows:

"The Khawarij who deviated, whom the Prophet (blessings and peace of Allah be upon him) enjoined us to fight, and whom Ameer al-Mu’mineen ‘Ali (may Allah be pleased with him), one of the Rightly-Guided Caliphs fought, and whom the leading scholars of Islam among the Sahabah, Tabi‘een and those who came after them were unanimously agreed upon fighting, were not described as disbelievers by ‘Ali ibn Abi Taalib, Sa‘d ibn Abi Waqas and others among the Sahabah; rather they regarded them as Muslims even though they fought them, and ‘Ali did not fight them until they shed blood unlawfully and raided the property of the Muslims; then he fought them in order to ward off their wrongdoing and aggression, not because they were disbelievers. Hence he did not take their womenfolk captive and he did not seize their wealth as booty." (Majmu'a al-Fatawa 3:282)

However his own actual position is in his work al-Nubuwwat and derived from when he quoted the words of the Holy Prophet (peace & blessings be upon him) himself: “’They shall exit from Islam just as an arrow exits a hunted game’. [Ibn Taymiyyah opined:] Their exiting from it means their leaving it due to their belief that it is lawful to shed the blood of Muslims and seize their wealth”. (pg. 225)

This is an opinion of a "large body of scholars" (Asqalani 12:313).