RSS certainly solves a real problem, so I think there is an opportunity for reinventing RSS with better UX. Don't ask me how though.
It isn't the protocol, it is that there is no profit in an open standard like that. Google killed their Reader because they wanted everyone forced into G+, rather than consuming content from around the Internet.
It is the same thing with messaging and VOIP. Nobody wants to use an open standard like XMPP or more recently Matrix because there is no money in using standards when you can lock users in to a walled garden.
Well, you need to have a feed reader set up.
That might be partially on the onus of the reader, but it would require you to at least install an extension to add their site as your default feed processor. That can be a UX loss.
I remember trying to get into RSS feeds a few years back. I passed them by - it was easier to just hit news.google.com or my multireddit or twitterfeed for news I was interested in. Sure, I didn't "get it," and yea fine I'm not the smartest guy ever, but I'm probably a good representation of the average net user - interested in efficient means at getting the news such as RSS, but also tremendously lazy. Why bother? I already have a twitter/reddit/gnews account.
I barely ever click on any RSS buttons. Partially because sites nowadays don't even bother putting one in there, partially because TinyTinyRSS extracts the feed from the site's <head> automatically.
So running analytics on that button would be very misleading, especially for users like me ;-)
Set up a wordpress site, you have RSS built in. Set up a drupal site, you have RSS built in. Set up on Squarespace and you have RSS built in.
Pretty much every off-the-shelf CMS has RSS with zero effort, you have to actively turn it off. It is only the people who put a bunch of work into building a site without a CMS that don't easily have RSS.
I don't want to live in the same internet as people who can't copy & paste an URL.