You can find the rss feeds of (almost) all podcasts in the world here: https://www.listennotes.com/
Or use the API: https://www.listennotes.com/api/
Or use the API: https://www.listennotes.com/api/
As the episodes are syndicated it seems reasonable to "rebroadcast" the episode on your site, but the terms and conditions for many podcasts (example 99 Percent Invisible) seem to suggest that is breaching their usage rights.
Have you looked into this at all and did you find similar issues?
How is the audience feedback so far?
June, 2018: https://www.producthunt.com/posts/listen-notes-3-0
Dec, 2017: https://news.ycombinator.com/item?id=15825900
But one comment that I heard again and again is: It's meaningless to build a podcast search engine, because Google and Apple can easily build something similar :)