Even if the library wasn’t burned down by Caesar, it may still have been burned. In the 4th Century CE, Emperor Theodosius outlawed pagan practices and ordered the Temple of Serapis in Alexandria to be burned. So it is possible that the Emperor also ordered the burning of the library, but there are no sources that reference a library burning or being destroyed.
The final time that it could have been destroyed was in 642 when Alexandria was captured by the Muslim army under Amr ibn al-As. There are Arabic sources that claim the Caliph Omar ordered the destruction of the library. However, these Arabic sources were written long after the destruction was said to have occurred, and some historians believe the writings were politically motivated.
A final possibility for the library is that it just fell into disrepair due to lack of interest, budget cuts, and the scrolls being moved elsewhere. As mentioned previously, the library was somewhat on a decline even before the supposed burning by Caesar. If the library was partially burned three different times, the scrolls and scholars may have left for other libraries.
There were several capital cities that had their own centers of academic excellence. The Imperial Library of Constantinople stood until 1204 and housed much of the knowledge and works of the Greeks and Romans. In fact, many of the Greek works that survive today are due to the Library of Constantinople. There was also the Academy of Gondeshapur which came later in the 6th and 7th centuries, but focused on promoting science and medicine. Part of the work of the academy was to translate the Greek and Syriac texts, and therefore many scientific and medical ancient texts survive due to this Academy.
There was also the House of Wisdom in Baghdad from the 9th to the 13th centuries and it held the largest selection of books in the world at the time. The House had observatories and centers for study of science, math, medicine, alchemy, chemistry, and geography. Much of the knowledge and study at the House of Wisdom drew on the Greek texts but also on Syriac, Indian, and Persian texts. Many of the texts that would have been destroyed at Alexandria could have had copies that were brought to the House of Wisdom.
There are things that may have been lost at Alexandria, but not nearly as much as some people believe. The works of Hero could have been there, some of them were saved by the Arabs but not every one. The works of Aristarchus of Samos could have been there as well the works of Hypatia. But the Library of Alexandria was not solely responsible for the loss of these works, and many of them would have been highly sought after around the world. This means that these works could still be out there, or were possibly destroyed multiple times as other libraries were destroyed by conquerors.
It’s also important to note that the remains of the library have yet to be found, so there is still a possibility one day of having answers to all of the mysterious questions of the Library of Alexandria.