I want to either set an alert on an RSS web part (tried, but it seems that it's only for the page and when the RSS changes, I get no alert) OR send the RSS content to a library or list as well. Key is for people be able to search the RSS content. Not a ton of people who've tried this, at least that came up in searching.

SharePoint Alerts are only generated on lists and libraries. RSS feeds are just a display of external content. There is no out of the box feature to sync or import RSS feeds. Best bets are:

• Write a SharePoint timer project that fetches the RSS feed on a schedule and updates a list.
• Write an external application and run it as a Windows scheduled job that reads and syncs the RSS feed to a list.
• Ask the users to add the RSS feed to their Outlook client or to a third party RSS feed reader program.