It’s only a baseless presupposition. The driving force that gave rise to the Islamic golden age was Islam itself. See the wiki link below under “Causes”
Islamic Golden Age - Wikipedia
at least. The translations from Greek into Arabic were done by Nestorian Christians.
The Islamic state embraced the Islamic principles of openness and tolerance, even so Muslims were the majority but non-Muslim minorities worked along side with Muslims and made their contributions especially with respect to translations to the Arabic language.
Most of the early math and science was done by non-muslims
Irrelevant, what is the point about early science?
The route of knowledge always starts by acquiring available earlier knowledge of others then you build on it. To get your PhD, first, you must acquire available knowledge of others. Second, you start your own research/inventions. There is no other way. Did you ever hear of a scholar who ignored all earlier knowledge of others and started his own knowledge from scratch?
You must start by getting the knowledge from available sources, from that point you may advance to the next level. This is exactly what the Muslims did.
(there were early restrictions on who could convert to Islam)
There was no such thing as restrictions on who could convert to Islam. It was up to the citizens of the unified territories to convert to Islam if they wish or pay taxes to the Islamic state, which is equivalent to the Muslim citizens obligation to pay zakat.
The elimination of the Persian empire and the (brief) unification of a large territory certainly had a lot to do with intellectual (and economic) growth under the early caliphates. What isn't so clear is the influence Islam itself had.
Remember that, at first, there weren't very many moslems in the Islamic lands. The elite were moslem, but the vast majority of the populace was not.
People of the unified territories were free to embrace Islam if they wish, at the time of the golden age the majority of the population were Muslims. The Islamic state embraced the Islamic principles of openness and tolerance; it was not about Arabs but rather about Islam and the Islamic state. The scholars of the Muslim world were both Arab and non-Arab, Muslims and non-Muslims, even so the elite including rulers, scholars, scientists were mostly Muslims but non-Muslim scholars worked along side with Muslims and made significant contributions especially with respect to translations of known classical knowledge of the world into Arabic.
See the link below. It provides knowledge about numerous scientists of the Islamic golden age. These Muslim scientists had overarching knowledge including multiple fields of science as well as Islamic theology.
List of scientists in medieval Islamic world - Wikipedia
See the link below for the article titled “Early Islamic philosophy”. Here is a quote:
“The period is known as the Islamic Golden Age, and the achievements of this period had a
crucial influence in the development of modern philosophy and science. For Renaissance Europe, "Muslim maritime, agricultural, and technological innovations, as well as much East Asian technology via the Muslim world,
made their way to western Europe in one of the largest technology transfers in world history.”
Early Islamic philosophy - Wikipedia