The books using the same provider use the same odds which means that you can’t arb. To arb the player uses books with different lines. Also when you arb you need to bet more than $200 on both sides to make it profitable. It is semantics in a way since value betting is the same argument for the book.
But you are right in that the book profiles the player, sets limits and the book gets that information. The OP gave us enough information in this thread to figure out what the book is looking at.
You are not wrong but I think you are not considering the odd changes.
Arbitrage betting needs patience. In an sport event odds are changing all the time. Going in favor of a side or against it depending on the information of the market. An arbitrage bettor waits for the changes and once it is in a good point for him to place another bet in the different side, they can easily use another sportsbook instead of the same one so that the same one can not blame him for arbitrage betting. It's much easier for the live match than the pre-match odd selection.
You aren’t going to make the most profit buying back in house. You use different books.