Hi there.
I'm not seeing any messed up code or headers or anything like that. As Gaston said, it might be after-effects of broken plugin or something.
At this point, i'd recommend this: do manual fetch as google request, see if it comes up. If not, disable all the plugins, do new fetch and see if it comes back, re-enable plugins, fetch, see what happens. this would help you troubleshoot.
I recommend doing these steps on development server - basically replicate your website on like a subdomain URL - that way you can do all the plugin disabling etc and do all the fetches without affecting currently ranking site. Also, after replication on dev server and doing fetch right away, you'll know if your home page is just being still cached by google in the state of noindex due to previous mess.
Also, make sure that in sitemaps you have last-modified set to as recent time as possible, otherwise google won't update the page index, since it's technically considered cached, if not modified since last crawl.
Hope this makes sense.
Yes, it's confusing to bots and seems kinda shady to me. If it's possible, do it straight forward - canonicalize to prefrred url.