It is illegal to use a mailbox, yes. (Not necessarily a mail slot). You are looking for 18 USC 1725.
Whoever knowingly and willfully deposits any mailable matter such as
statements of accounts, circulars, sale bills, or other like matter,
on which no postage has been paid, in any letter box established,
approved, or accepted by the Postal Service for the receipt or
delivery of mail matter on any mail route with intent to avoid payment
of lawful postage thereon, shall for each such offense be fined under
As to why the monopoly exists, that's a legislative history question. There are all of the obvious advantages and disadvantages: network effects, interference with official mail, government income, freedom from spam; inconvenience, making it easier to receive unofficial mail, increased competition in mail delivery, no need for a second mailbox for newspapers, better advertising information from said spam.