Would you be willing to address the history of "agent being paid by publisher, agent sending check to writer?" Why is it not the other way around, writer paying agent?
Mostly cause in the olden days, when this protocol began, authors owed money to their agents. Authors were broke, borrowed money from agents (remember this is the days before Visa Mastercard and AmEx were glad to lend you cash at exorbitant rates) and agents recouped costs from advances and royalties...most of the time. For a narrative of how that worked, just take a look at the biography of Richard Yates, A TRAGIC HONESTY by Blake Bailey. Here's a link to an interview about the book on Identity Theory
Plus it makes sense to have an agent watching your royalties. If you've got a publisher with a screwy accounting system (ie ALL of them) an agent has dealt with it before. You don't want to be arguing about whether the amount is right after you cash the check, trust me.
And an agent generally has more leverage with a publisher. You're one person. Even if you're doing ok, the agent has more leverage cause s/he's got several authors with the publisher.
And keeping track of what you're owed, and when, is one of those details that writers tend to be BAD at.
The system evolved into what we have now cause it made sense.