Does an attorney automatically owe a duty to someone who pays the legal bills of that attorney?

No.  Consider, for example, situations when a parent hires an attorney for their child, and pays that attorney's legal bills. The child is the client of the attorney, not the parent.  The attorney owes a duty to his/her client, not necessarily  the party or parties who are paying the bills for that client.  If the rule were any different, that might invariably lead the attorney into an unavoidable conflict of interest.