Already doing this. I use gPodder, download the videos/podcasts to a directory Plex is watching, and its done!
I currently use it for downloading subscribed videos from Youtube, and it works perfectly. Its cross-platform too.
You can grab podcasts through rss feeds via a podcast client such as this: gpodder. You have to go through each website and get the rss feed page and then add it through the url link option in the menu IIRC. Here's JRE's RSS feed page.
Or if you want you can just go to the podcast page of joerogan.net and right-click the download link and 'save link as' if, for whatever reason, you have not thought of that.
Podcasts are probably a better deal for variety and convenience. Set up a laptop with something like gPodder and subscribe to a lot of shows. Drive into town for the nearest WiFi for your supplemental groceries etc and download your subscriptions.
Listen to them whenever you like and you also get the "TiVo" benefits of pause, rewind and multiple listenings—and no need to fiddle with the radio at odd hours of the day and night to AM DX. You're on a homestead for crying outloud—aren't you tired!? ;)
I use a video podcatcher. I personally use gPodder, but there are plenty around.
If you click the CAEN Lecture Recording CTools page, there is a link for the Video podcast. Just add that link to the podcatcher tool of your choice. Many (including gPodder) will download new videos for you, and keep track of when new videos are available.
RSS stands for Really Simple Syndication. Think of it as a blog, but in a specialized format. To use RSS you have to have an RSS reader. There are web based ones (such as Google Reader), desktop-based ones (such as Feedreader), and some specialized ones (such as gPodder) for podcasts and the like. If you subscribe to podcasts on iTunes, you are using an RSS feed.
Die XML Datei der Bohnen ist ein ganz normaler iTunes kompatibler RSS Feed. Jeder halbwegs brauchbare Podcatcher sollte damit zurecht kommen, mein Instacast auf Mac und iOS hatte keine Probleme.
Versuch doch mal gPodder, das ist open source und läuft auch auf Windows.
I use gPodder to download the episodes combined with [insert your media player of choice here] to play them. I like gPodder because it's functional and streamlined. I can only think of a couple downsides, both of which may be entirely irrelevant to you:
I haven't found anything great like there is on mobile, but I make do with gPodder.
It handles the basics -- subscribing, downloading, streaming (albeit in another media player of your choice). Nothing special, nothing great.
For windows? If you listen to podcasts a lot, gpodder.
Note that I personally only use gpodder for a handful podcasts with broken feeds. iTunes is just much more convenient with the user reviews/ratings and built in player with auto resume.
Imho the current UI is also much better than it used to be (but certainly still wonky, e.g. scrolling through episodes absolutely sucks and software engineers at Apple should be absolutely ashamed for shipping an UI this laggy).
If you're looking for desktop software (since all the responses have been mobile apps) I found a few resources you might want to check out.
http://www.makeuseof.com/tag/what-is-the-best-podcast-manager-for-windows/
This article is from 2013 but most of those are still relevant today.
The only one not on that list was gPodder which seems to be popular.
As far as content goes, this site seems to be a decent start :: https://gpodder.net/directory/
You can probably google 'Podcast directories' to find some more stuff.
I had a bad string of mobiles whose headphone jacks would fail in various and spectacular ways, so for a very long time I used a teeny little Woot-purchased MP3 player for listening to podcasts. While I have moved on to and also very highly recommend BeyondPod (your fears of battery loss are unfounded), my old process was to podcatch and then upload them directly to the device. I tried a few different podcatchers (I though Juice was a faboo interface but kept having a software fault) and eventually settled on gPodder.
In your case you could set up a Dropbox for your save directory, then install the Dropbox app onto your device and use it to download the actual files when you want to listen.
I've had my Nexus 5 running all day with every-six-hours checks for new podcasts and that app isn't even registering in the battery use settings screen, so I'll again suggest that using an on-device app is your simpler bet. If you want a two-piece solution, good luck!
It's what I use for "podcasts" that are YouTube only. You have a to dig a bit to find the exact "feed" and quality settings you might need, but it currently downloads the video files from YouTube so I can watch them offline and/or at 2x speed as I prefer.
YouTube could change its code tomorrow and gpodder might stop working, but it's worked for the past couple of years for me right now.
Of course I figure Tom, once the dust has settled, will probably release his own podcast feed and maybe even a Patreon subscription or something along those lines.
As the image indicates, I only selected iTunes+quicktime. Is it important? Probably not.
Does it help? Not at all. I'm all in favor of open source software. I usually use gpodder as my podcast client; but I've recently started using Audible for audio books and used iTunes for the sake of simplicity. I'm sure, in time, I'll find a replacement to iTunes with Audible.