Old hosts die hard in public radio.

When 73-year-old Garrison Keillor retires as host of “A Prairie Home Companion” next month, he’ll leave more than 3 million weekly listeners loyal to a show that began more than 40 years ago. Elsewhere on the dial, “Car Talk” ranks near the top of National Public Radio’s ratings even though co-host Tom Magliozzi died at age 77 nearly two years ago—his jovial cackle still echoing in “best of” versions of the show on more than 600 stations nationwide. Later this year, Washington talk-show...