I would recommend looking at the raw xml of those sites. There is no information about what you want. Most rss readers display the article/episode and allow you to read it or to save it.

You would likely need to combine the RSS feed articles with data provided by a service that scrapes the website in order to identify where the article is placed at any one point. Sounds doable even if scraping is always fraught with numerous pitfalls. I haven't heard of any such solution, and building one is not on my todo-list, sorry.

Newsblur allegedly allows you to "train" it to suggest certain stories and ignore others, but I've never used that particular functionality.

I haven't seen any information that could be used for this in the RSS feeds I've looked at. You could scrape the website, especially if it's all running on your own computer, but if you do it on a server you'll almost certainly be blocked unless you use a third-party scraping service. The WSJ in particular is super aggressive; you'll probably be OK with the NYT, which has a personal use exemption.

Unfortunately Anthropic and OpenAI have kind of ruined scraping for everyone else.