I understand the purpose of the bolt-action, semi-auto, and full-auto rifles but don't see the purpose of the lever-action except for novelty.
I'm not even sure I'm following the logic of the question? What is the purpose of any firearm? It's for whatever purpose the owner chooses.
Hunting? People have and do hunt with bolt actions, semi-autos, pumps, break actions, lever actions, rolling blocks, falling blocks, and black powder muskets. They all work, and one person's choice is just that, their choice. There were plenty of bolt action and other alternatives back in the peak days of lever guns popularity, just as there are now.
Personal or home defense? Same thing applies. People have been defending them and theirs with every type of firearm ever created. It hardly needs be an EBR, battle rifle or modern conception firearm to be an effective home defense weapon. Again, what one person chooses versus another is just that - a matter or personal choice or preference.
Recreation? Lets be honest, this is a huge reason why most people who own more than one or two firearms own as many as they do. Speaking for myself, I bought my 1892's mainly because they are just great fun to shoot. In a pinch, they'd also make a decent HD closet gun, they are also great general purpose woods guns for me (an 1892 has to be one of the most comfortable long guns to carry around that I know of - it just falls to the hand perfectly).
Seems to me you could ask this same kind of question about any type of firearm that does not appeal to you or which you have no desire to own. What is the purpose of black powder rifles? Or single shot hand guns? Why does anyone buy a Ruger No. 1 anymore, or a coach gun?
I say thank goodness there is so much variety available, and that so many of the classic designs that have stood the test of time are still being made. I'll be hanging on to my lever guns, 1911's and revolvers until the day I die, and I'll go with a big smile on my face too.