The rabbinic tradition preserves both sides of the judgment on Bar Kokhba, and the preservation is itself the point. Akiva's acclamation is recorded, and so is the reply attributed to Yohanan ben Torta, that grass would grow from Akiva's jaw before the son of David came. The tradition kept the sceptical line, and history vindicated it.
The effect on Jewish thought was long. Messianic expectation did not disappear, but the idea that it might be brought about by armed action against an empire became suspect. Talmudic passages warn against forcing the end and against rebelling against the nations, and the pattern in later centuries is that messianic movements are met by rabbinic authorities with caution or open hostility rather than with enthusiasm. Two catastrophic wars in seventy years supplied the reason.
The other legacy is martyrological. The persecution that followed the revolt, including restrictions on Torah study and on circumcision, produced the tradition of the Ten Martyrs, with Akiva among them, whose deaths are recited in the liturgy of the Day of Atonement and of Tisha B'Av. A defeat that ended Jewish political existence in the land for eighteen centuries entered the prayer book not as a military disaster but as a record of people who kept teaching when teaching was a capital offence.