Here's some answers to some of your questions based on my personal experience over the years:
What was the original purpose of the Home button?
home button stems from the time when the internet was a new thing to home computer users. Being a short link to the browser's default start page, it offers a way of consuming content and as such it was more comprehendable at a time when browsing interlinked media was not yet common.
Just like the front page of a common printed newspaper, front pages often were content aggregators provided by browser manufacturers (msn.com for internet explorer or apple.com for safari, for example) or teleoperators. The front page was, historically, a common point of entry for the browsing experience. The
home button offered a way to return to a familiar web page that offered a collection of links to go to next as well as a familiar "surrounding".
home page also refered to people's own web sites. Before the commercialisation of the internet, it was common place for people to have their own personalized web sites (much like social media profiles nowadays), which they also accessed as a point of origin of a browsing experience, with own content publishing interfaces and link collections. Thus folks used to also set their own personalized web pages to the
home button. (Relying mostly on personal experience here with this argumentation, but at least the "home page" site on wikipedia seems to somewhat agree.)
Was it good UX to have that button or was it useless to begin with?
Personally, I think it was a good user experience. Just like the
back button, the
home button was one of those really reliable escape routes for the user - and in the early wild, wild days of the internet you'd sometimes just want back to the start of your browsing experience, so to say.
Is it still relevant in today's usage of browsers?
If not, for what purpose is it being retained?
Today, I think the
home button suffers a tolerated existance for the sake of many users that have gotten used to using it over the years, but it's importance in browser user interfaces has been in decline. Personally, I don't see it in on my Firefox bar, nor on the Chrome - that might be attributed to customized interfaces, but I do think also the standard distributions leave the home button away.
What remains current is the functionality the
home button once offered, which was a shortlink to the beginning of browsing the web. Today, this just happens by opening a tab, and many browsers have adopted to showing a collection of most used websites as the "start page" - because users today are using the internet with more distributed points of entry they choose themselves, rather than through a start page with predetermined routes to content they like.