Replace Google/Feedburner with something else?

Currently using the “BFA subscribe” widget and Feedburner option in BFA Atahualpa theme. Like so many others, I am unable to reach my Feedburner account since Google took it over, and would like to regain a little control of feeds.

Main need is for people to be able to subscribe to several blogs (WordPress blogware, hosted on various domains) by e-mail or by RSS feed browser notification – nothing fancier than that. I would like to be able to collect a mailing list to see who’s reading; full statistics would be nice but not necessary. These blogs have only a few readers – not thousands – and only get updated every few days.

There are several plugins that seem to do the same thing, but none seems to have many downloads or ratings. There are also numerous web pages and forum posts suggesting services like Mailchimp (currently free at my use level) and Feedblitz, as well as WordPress.

Thought I understood the basic idea of feeds, but more recent trends like aggregation and connection to social media (not of interest for these blogs) make it all harder to follow. Can anyone here please point to a simple primer on getting basic feed services into WordPress? And will deleting the BFA Subscribe plugin and Atahualpa theme Feedburner settings effective disable the Feedburner account, or will it remain forever able to scrape these sites?

Thanks for any help.