But why? I don't see a financial motivator here, unless there is some long-game that I'm not aware of. In theory, driving users to websites in a search result is likely just going to make Google more money from displaying ads.
The only thing I can currently think of is they actually are trying to improve the user experience.
If you listen via Google they not only know what you listened to but for how long you listened, which parts you skipped over or stopped listening at, and whether you listened to the ads or not. All that information has value to them. Being able to say these people googled your product within 72 hours of hearing your ad on these podcasts is very valuable.
Maybe they indirectly want to 'improve user experience' (so you're less likely to leave) but it's not their terminal value.
Why would it be more profitable to send users to the BBC website?
This is the company that decides shades of blue by conversion numbers.
1. You see a podcast you're interested in
2. You go to the BBC's website
3. You discover you need to download BBC Sounds (if in the UK)
4. You download the app
5. You register with a username and password (and UK postcode, which narrows you down to the street where you live)
6. You press play and start listening
or
1. You see a podcast you're interested in
2. You press play and start listening
Wouldn't anyone prefer #2? Including the BBC's programme makers?
I'm a non-UK listener and use the BBC Sounds app all the time. The only thing that's unavailable are program downloads.
Unless you're side-loading on Android, which is perfectly possible too, but that's not quite "being available".