Can't find episodes even though does validate and has episodes

It's possible. www.youtube.com/watch?v=mTQNTm6-tFI Though how do I do it through SS? Feedburner seems to somehow do the magic... What you think about that?

I think the trick is podsync.net, which takes the youtube videos and creates an mp4 file from it.
At least as far as I understood it. Feedburner probably lets you customise the feed afterwards.

But that's not something SermonSpeaker can offer.

Thomas, thanks for all your patience. If iTunes won't work, what would suggestions be to our site visitors, how can we help them use podcast functions of site, if they want to subscribe to the video podcast? Since your component does have video option, I guess you'd have some ideas on how people can use it?

I probably totally do not understand the way podcasting does work. I was under impression that podcasting is just a convenient way to provide automatic downloads for regularly provided media content. iTunes here is nothing but a marketing tool that will present the feed link to anybody looking. All the rest is done by the program that I use for listening/watching.
I had been listening and watching podcasts for years. Now decided to provide this convenient way to our site's visitors. But seems to be I am too naive. I've learned now that iTunes (as all stuff from Apple) will jerk up when something is not their way. Well, I've manually added the video podcast to my phone to testrun the whole thing. And here's what we have:
yadi.sk/i/G1Te2bC63LCGtm - videoversion does find episodes, even takes time from settings. Though it won't read the filesize and won't play, failing with error. yadi.sk/i/vIi-4IV03LCHLk (tried both Vimeo and YouTube). Of course iTunes on my PC won't get episodes even when videopodcast is added manually, yet will see the feed was updated - yadi.sk/i/42XFLf3l3LCHfy . Though the audioversion of the same does work (files are stored on AWS): yadi.sk/i/HBffPBrL3LCHzc
At the same time while audioversion does work in iTunes, on the phone it will be found (I guess it is found through iTunes PodcastConnect, I doubt it searches all the sites of the world and finds it on my site). Yet, no episodes... yadi.sk/i/ZmU6i2CT3LCKHJ
So, I am totally lost. Could you, please explain how the whole thing with podcasts is working? What happens between storage, my site, and users device? What does Podcast Connect do in this lineup?

Technically, a podcast is just an RSS feed (and XML file) with some media attached.
If that media is downloadable, a feed reader like iTunes can download it and you can listen/watch it on the go.
If that media isn't downloadable, it depends on the feed reader what it does with the information. iTunes as you said will refuse to do anything because the feed isn't following the "Apple" way. Other feed readers may just offer the link to Youtube or Vimeo to watch the episode.
Some browsers have feed readers integrated or allow to add plugins for that. They can then subscribe to a feed directly from a sermon list (there is a feed link in the header of those pages).

Podcast Connect is just an Apple owned archive where you can register your feed. It allows iTunes users to search and find your podcast. That's all. Your podcast works fine also without you registering it at Podcast Connect. It's just a convenient way for Apple/iTunes users.

So if you really want to do a video podcast which is easy for the people to subscribe, you will have to save mp4 files and store them on AWS or your own server. I'm sorry, but Vimeo and Youtube will not work well for that. On the other hand, Youtube (and probably Vimeo) allow to create playlists which can be used for a similar thing like a podcast. That works without SermonSpeaker at all.

Hello Thomas, thanks for all your help. I've tried feeds from YouTube, podcast catchers don't work with them. But I've got one solution that does work for now, if somebody would need it - podsync.net/ - will create podcastable feed from YouTube channel the one that even Podcast Connect will work with.