I can’t find anything that specifically says a carrier must pick up regular outgoing mail, but for Priority Mail they must:
“Free, regardless of the number of packages you’re sending.
Your carrier will pick up your package when your regular mail is delivered.”
So if it’s worth paying around $4.00 per letter for faster delivery and pickup from your mailbox, go to your post office and order (free) Priority Mail envelopes and stamps. I think they can be ordered online too, with free delivery.
On this page, http://pe.usps.gov/text/DMM300/print/508.htm for rural mailboxes it says:
“Generally, mailable matter is collected from a rural mailbox if postage is fully prepaid or money equal to the required postage is left in the mailbox. Money in a rural box is left at the customer’s risk. When postage or money is insufficient to cover postage, the mail is not collected, or if the sender cannot be identified, the mail is treated as unpaid mail. Mailable matter not bearing postage found in, placed on, attached to, supported by, or hanging from rural boxes is handled under the applicable standards.”
I still say the bosses of your carrier should ensure he picks up whatever properly stamped mail you leave in your box, and if they don’t keep calling every day.