Islamic tradition holds that Muhammad was born in 570, the "Year of the Elephant", shortly after a failed Axumite campaign from Himyar on Mecca, led by one Abraha the Abyssinian. Modern historians dispute this, however, noting that from c. 565, Sassanid Persia governed Himyar and Abraha possibly died even before then. In light of this muddled timeline, when was Muhammad likely born?
Kaleb, the Aksumite king, following his conquest of South Arabia (Yemen) in the sixth-century, for whatever reason, did not change the local political structure. Rather than causing a dramatic change, he installed a Himyarite prince to serve as a client in the early 520s. The Himyarite prince was overthrown by the chief of the Aksumite army, Abraha, around early 530s.
Abraha did not rule on behalf of Aksum. He was an independent ruler. he styled himself as a Himyarite king, and seems to have viewed himself as continuing the rule of Himyar, not Aksum. His reign lated for nearly 30 years. According to Christian Robin, in Arabs and Empires, Abraha ruled until around 565. But the year in which Abraha died is not known in a definitive way. The last known inscription of Abraha dates to 559/560, he was succeeded by his two sons. Their rule, unlike their father's, did not last long.
As for the elephant-led campaign, the year 570, as known in Muslim sources, is not beyond the realm of possibility.
With regards to the Persian involvement, in Arabia and the Arabs, Hoyland mentions that the Persians got involved in response to a Himyarite plea for help to resist the Abyssinian invaders. The Persians sent a force, defeated the Abyssinians and installed a Himyarite prince, as their client. History repeated itself, the Himyarite prince was overthrown by some Abyssnians, at which point the Persians involvement became more direct, and Yemen was incorporated as a Sassanid province. Bowersock in The Throne of Adulis dates the start of the Persian rule over Yemen to 575 565.
As for your question regarding Muhammad's birth, 560-570 is a reasonable timeframe, and the traditional Muslim narrative, as far as I can tell, is not at odds with the studied and published South Arabian inscriptions.