If you;ve been messing with wordpress - I'd use that - especially if you can develop locally and use one of the 'static' generator plugins - you can update the blog, and spit out a bunch of html / css files - then you can ftp/host a lot of places real cheap and some free - I've heard git pages is an option somehow.
When you convert WP to static you don't need php or many other things running for the server, and apache or whatever can serve the files faster / not have to build them from all the wp and template parts with each request - so a cheap $5 server may be fine.
Of course when you spit out static wordpress you lose the ability for users to add comments to the blog directly, or contact forms, search functions and other such dynamic things..
but if the main goal is to blog or publish and cheap - I think it's a great option - you can always put an email addy on a contact page and tell people if they want to add a public comment to just email you..
this also has the huge advantage of being way more secure - no logins and exploits to worry about.
Not sure why you'd need something like cloudflare, unless you are super focused on the privacy / discovery / harder to hack.. if that's the case watch any kind of email setup or mx records some server setups come with.. surely there are other more private proxy services.
analytics - many cheap shared hosting options with cpanel come with awstats, analog stats, and a couple / few others you can check a box and have them log for you on the server.
I've seen a few others around HN the past year or so, but my bookmarks are unwieldy and I'm sure others can chime in on those faster.
If you are really looking for privacy first all around, probably need to be careful about where you setup domain name registering, and make sure the whois is hidden upon registration.