The stations CBS OWNS have to run it (and many of them didn't want to give the time back to the network).
The fact that some affiliates opt out of all of part of the program is telling.
CBS has always had a problem with clearances for its morning programs (except for Captain), even more back when it only owned five stations. It may be just as well that much of the country doesn't remember Walter Cronkite doing cigarette commercials and talking with a lion puppet on an early version of the morning show.
I still miss the CBS Morning News with Hughes Rudd, probably the best regular network news broadcast ever (possibly excepting NBC's Overnight news show with Linda Ellerbee and Lloyd Dobbins). Of course, most of the country also got to miss the train wreak CBS created when the brought a completely inexperienced Sally Quinn on as co-anchor (to promptly fall flat on her face).
But the current Morning Show will likely not sleep with the fishes as long as one of the girls on the show sleeps with Les Moonves.
Here is CBS' current station portfolio.
New York, NY | WCBS
Los Angeles, CA | KCBS
Chicago, IL | WBBM
Philadelphia, PA | KYW
San Francisco, CA | KPIX
Boston, MA | WBZ4
Dallas, TX | KTVT
Detroit, MI | WWJ
Minneapolis, MN | WCCO
Miami/Ft. Lauderdale, FL | WFOR
Denver, CO | KCNC
Sacramento, CA | KOVR
Pittsburgh, PA | KDKA
Baltimore, MD | WJZ
Salt Lake City, UT | KUTV /St. George, UT | KUSG
Austin, TX | KEYE
Green Bay/Appleton, WI | WFRV/Escanaba, MI | WJMN