How much would it have cost to attend the Beth-Safar?
Who else would pay the teachers?
So you reckon only rich people became Rabbis?
I'm sure there were quite a few itinerant rabbis not attached to the official schools, just as there were lots of "faith healers" and peddlers of religion all over the Roman empire. But as it would cost money (and take children away from earning from a crust for their family), yes, the more educated you were, the more privelaged you would be. There may have been some patronage of the poor by the rich, but not much.
This would be true right up to the provision of universal education, by the way. Until the invention of printing, when access to the written word became more and more affordable, most people would never see books. Parchment was expensive, paper unknown until the middle ages, scrolls were unwieldy and books as we know them (in codex form) were actually an invention of the early Christian period. Imagine how long it would take to copy, meticulously, word for word, a copy of, say, the Book of Genesis. Many professional scribes were themselves only functionally literate. Scrolls and manuscripts of the scriptures are littered with mistakes, deletions and erasures.
We're talking about a completely different world. An oral world, not a literate world. There is no evidence of universal schooling in any part of the Roman empire.
In a highly stratified society like the ancient world, nobody cared for the poor or slaves. They were the bottom of the heap, they did they dirty work so that the rich could dispute theological points in synagogues.
It's highly unlikely later practices like the Beth-Safar were being practised with the kind of rigour that they may later have been practised. Look at England even a couple of hundred years ago; there were more readers per head of population then than in the ancient world.
The Bible was not written to be read in the privacy of one's home. It was written to be spoken aloud, in the Temple or in the church.