I sent out a book last week via first class mail. That's what PBS wrapper printed and at PO they said that yes it was

cheaper than media mail due to weight of book.  Today I noticed that the expected receipt date is the same as if I

had mailed it media mail.  Does the system not track  how the wrapper is printed??  That is first class vs media mail?

The system will put the same amount of days until the "lost" date regardless of how it is mailed.  However, you can expect that a book mailed first class will most likely get there before a book mailed media mail.