Also, no, this is not the message in the Gospel. Here is an example commentary of the scene if you actually want some context: https://biblehub.com/commentaries/ellicott/matthew/15.htm
Citation needed.
Pretty much all secular scholarship I have ever seen on this topic has concluded that both rabbinic Judaism and early Christianity were reactions to the destruction of the second temple. Rabbinic Judaism is the direct intellectual and religious descendant of the pharisee sect, and has always stood on its own merits. The only rabbinic tradition that definitively developed in response to the rise of Christianity is the birkat haminim, a prayer that was used to "out" secret Jewish Christians before the final split between the religions. Beyond that, all there is are attempts to deal with / respond to persecution of Jews and Jewish institutions by Christians in later centuries (e.g. the fixing of the Jewish calendar after Christian Rome disbanded the Sanhedrin).
Don't overstate the importance of Christianity to the early rabbis. Rabbinic literature is generally dismissive of non-Jewish religions and Jewish Christians like the Ebionites were seen as heretics -- a term the rabbis used for non-rabbinic Jewish movements of the Roman era (of which there were many, none of which have survived).
As for the Gospel, it does not really matter what people thought hand washing was for -- ritual washing hands did improve their hygiene, just like ritual bathing. Jesus told his followers not to bother, and it does not matter whether or not he understood the hygiene implications, because not washing their hands was still detrimental to hygiene.
The Scripture is full of similar scenes. Peter sees a vision with a sheet with unclean animals ready for eating. The focus was on evangelizing the people who eat those foods tho: the gentiles.
The destruction of the Second Temple was prophesied by Christ, and apostles started serving in the Church only 50 days after His resurrection, on Pentecost. So yes, rabbinical judaism has been a reaction to the destruction of the second temple, but the Church was already active and persecuted in the center of the roman empire, many epistles were already written and there were churches through greece and rome
That line of thought will not lead to a good place.
Really though, what makes you think any form of Second Temple Judaism could have survived the destruction of the Second Temple? Moreover, if Jews could find a way to keep Judaism alive in some form after the destruction of the First Temple, why should it be surprising that some form of Judaism was kept alive after the destruction of the Second Temple?
Obviously rabbinical judaism believers see themselves as the actual following: and that's where we disagree. However their explanation seems stranger than the christian one: we stopped having major prophets because the Son of God Himself came to reveal the Father and to save us, we don't need animal sacrifices and a Temple from human hands, because they were just a shadow to the Cross, prophesies from OT were fulfilled with the promised Messiah. The rabbinical judaism: they don't have prophets, because ..? Why would God let their temple be destroyed, if they were right about Jesus? Where is their messiah?
Islam obviously disagrees.
You asked where the Jewish messiah is, but like the old joke goes, for Jews the problem is waiting and waiting for a messiah that has yet to come, but for Christians the problem is that the messiah came and the world did not change.
Obviously rabbinic answers are not going to be very satisfactory or convincing to you; if rabbinic teachings made sense to you then you would have converted to Judaism by now. That said, rabbinic teachings are not a secret, so here are some answers to your challenges:
The animal sacrifice system was, according to rabbinic teaching, sufficient but not necessary to atone for sin. Among other things Jews point out that during the Babylonian exile there was no temple in Jerusalem and no sacrifices being offered, but that God did send prophets to explain to the Jewish people how they can deal with sin while in exile and without sacrifices. Another point is that when the sacrificial laws are stated, non-animal sacrifices are explicitly described; poor people could, for example, offer just a handful of flour if they could afford nothing better.
Why did God let the temple be destroyed? For the same reason the First Temple was destroyed. The Jewish people were punished for their national sins. In the case of the Second Temple, part of the national sin was dividing into competing factions who would sometimes try to use the oppressive Romans as a means to undermine competing sects.
Why are there no more prophets? In fact the rabbis do not have a single answer, it remains unclear in rabbinic tradition. Is that a problem? There are plenty of unclear, unanswered questions for Christians too; why is Jesus taking so long to come back? That said, some rabbinic answers include: the Jewish people are being punished for national sins; the true end of prophecy was the destruction of the First Temple, with subsequent prophets only repeating earlier prophecies; the prophets were sent at a time when the temptation to turn to idolatry was particularly strong, but that temptation diminished when the Jews returned to Israel and began building the Second Temple.
Maybe none of that works for you, and that's fine -- Jews are not actively seeking converts, so the opinion of non-Jews does not really matter; the Jewish view is that Judaism is for Jews and that the rest of the world only needs to hold to a handful of common sense morals (like not murdering each other and settling disputes in peaceful courts; I assume that much makes sense to you).