I would agree it's the researcher's responsibility. It's not that the company put up a webshell for kicks. The researcher found an exploit (good), and used it to install a webshell, demonstrating the highest possible risk (fine).
Once the shell is up, anyone who finds the URL has code execution on the server, because that's what a webshell is. Using it is a different skill than installing it.
Imagine I figure out how to jackpot your bank's ATMs, and I demonstrate this by setting a public ATM into "press button to receive $20" mode, pressing the button, getting $20, and sending you a letter describing how I did that, with the $20 scrupulously enclosed. Meanwhile, the ATM remains in the state of "press button to receive $20". How happy would you be?
Was it publicly discoverable?
Technically, yes, though realistically you'd have to guess the URL. I would find it pretty funny if one attacker got access somewhere by guessing the URL of a webshell installed by a different, more self-sufficient attacker, but that's not to say it doesn't happen.
Was it publicly exploitable?
Yes; the researcher didn't set up any authentication or anything.
I.. just can't wrap my head around that.
Once the notification is in and the shell demostrating it is up it should be immediate redeploy to a clean state, fix the hole, redeploy to a patched state.
The shell disappears on step one.
Instead some moron has the audacity to get all hurt because the broken system he is responsible for has not been patched back by the attackers?
What is this lunacy?
If the URL was unpublished, isn't that the same-ish as password protected?
All about bits of entropy i.e. difficulty if guessing.