Let me start with your statement that there is no rabbinic use of Isaiah 7:14 as a messianic prophecy. This is definitely not correct. For example, Hillel, the most respected first century Rabbi, used Isaiah 7:14 as a messianic text. This is found in the Targum Jonathan. Hillel noted the connection between Isaiah 7:14 and Isaiah 9:6. He saw the clear messianic nature of Isaiah 9:6, and concluded that Isaiah 7:14 is also messianic. He believed that it pointed to Hezekiah as Messiah. Perhaps he did this to oppose the Christian application of the two passages to Jesus of Nazareth, or perhaps he sincerely believed that Hezekiah was the Messiah. Either way, we have proof here that leading Jews in the time of Christ considered Isaiah 7:14 to be Messianic. If anyone says the Jews never considered Isaiah 7:14 to be messianic, they are mistaken!
I would argue that we also have solid evidence (but perhaps not so strong as to call it proof) that Isaiah 7:14 was considered messianic before 200 BC. This comes from the Septuagint–the Greek translation of the Hebrew Bible made in Alexandria around 220 BC. In this translation, the Hebrew ha almah is translated as the unambiguous Greek word for a virgin. In other words, the Jewish translators saw that this was a miraculous event, and concluded, therefore, that the birth of the child was to a virgin. That this makes the passage a messianic one is not a big leap at all. I believe that the reason the Jews translated the Hebrew for young woman as virgin in Greek was for the rather obvious reason that Isaiah 7:14 calls the birth a sign (ie miracle) from God. The same reason that the early Jews saw this passage as implying a virgin birth, and messianic is the reason I see it as implying a virgin birth and messianic as well.
Scholars have three options (or perhaps more, but these are the three most likely) for Isaiah 7:14. Either it is a prophecy of the birth of Hezekiah, it is a prophecy of the birth of the Messiah/Jesus of Nazareth, or it is a double-prophecy–in other words, it is a prophecy of the birth of Hezekiah as a prefigure and of Jesus as the thing prefigured. I believe that either the second or the third are more likely than the first because, as far as I know, no one considers the birth of Hezekiah as being from a virgin, or in any real way miraculous. This is a disputable matter, and honest, sincere believers will have different opinions about Isaiah 7:14. However, my thought is that this is a double-prophecy about Hezekiak as a type and Jesus of Nazareth as an antitype. To me, given the miraculous nature of the birth and given the rather obvious connection to Isaiah 9:6, I believe that it is pretty much of a slam dunk that this is a messianic prophecy. It is encouraging to note that Jews in the first century saw it this way as well.
From there, you pass on to the believability of the actual virgin birth. Was Mary indeed a virgin? Might she have lied? Well, one can argue that we only have Mary’s word on this, but that is not exactly true because we also have Joseph’s word that the angel told him Mary was with child (presumably as a virgin). Here is what I say: Look at the life of Jesus. He lived a sinless life. He fulfilled dozens of messianic prophecies beside Isaiah 7:14, he walked on water, he calmed a great storm, he healed thousands of lame, blind and deaf, he raised more than one person from the dead, and he himself was raised from the dead. It makes absolutely ZERO sense that this same Son of God was born from a deceitful mother who covered up her adultery. I am using logic and common sense here, but I believe that common sense definitely applies in this case.
Here is a source on Jewish interpretation of Isaiah 7:14. It is very thorough. https://www.ewtn.com/catholicism/library/problem-of-isaiah-714-12322