So what should they use for bumpers?
Since most stations run syndicated programming in automation, do they need bumpers anymore?
Just do a three count for stations joining late, and start talking. Local shows certainly don't need bumpers, especially with devices that can delay the top of the hour network news by a few seconds if necessary.
If that doesn't work, pick music not likely to be heard on similar shows so that people associate it with YOUR show, not everything else out there. Or pick music that thematically fits the show topic (a lot harder and probably impossible day to day). Or hire a producer and create your own (expensive, I know, but with all the restrictions on licensed music for podcasts and the possibility of artist fees in the USA, it's worth considering on a syndicator's budget). Just please don't buy a package from Company XXX that makes jingle clones that haven't freshened since 1988.