The existence of the priesthood and the temple did not diminish the ability of Jews to pray to God directly. Look at all the mention in the gospels to the synagogues. They were houses of study, worship and prayer. These synagogues were not run by the priesthood.
After the giving of the Tabernacle, did God ever accept sacrifices done by any-old-body done any-old-where?
Just because God allowed their existence doesn't mean that He blessed the goings on. He never commanded the Synagogues to be set up, He did however command the building of the Tabernacle, and the commissioning of the family of Aaron to act as priests.
I think the synagogues served a useful role. In every major city there was a synagogue full of people who knew the Scriptures. These acted as starting points for Paul's missionary activities in the cities he visited.
Didn't the synagogue get its start from the Jews of the Babylonian Captivity? For some reason I remember reading this somewhere, maybe in Edersheim's Life and Teachings of Jesus the Messiah.
-ksen