It seems to only screw things up as I try to type.
Way back when, the “Insert” key toggled between overwrite mode and insert mode. Insert added the letters in between the existing letters, wherever the cursor was. Overwrite would replace letters instead (existing letters were deleted as new ones were typed).
Nowadays, few programs listen to the insert/overwrite button at all and use insert as a permanent default.
In the 3D image program, Maya, pressing the Insert key lets you manipulate your objects pivot point. XSI/Softimage uses the Insert key as the “Split Edge” tool, also useful.
For word processing, it lets you type over things, which I’m sure you know. I’ve never used that feature personally, so I, like you, often find it frustrating.
However, when I was a radio DJ, we had a music and commercial scheduling system that took both mouse and keyboard input. I always used the “insert” key to insert a song or spot at a certain place.
Still remember the sequence: Insert, F8, Enter, Enter 
It made sense in the 80s.
It’s the reason you don’t need to use whiteout on the monitor and then retype everything. 
It lets you go back and add text prior to the current cursor position, among other things. Some applications will have specialty functions assigned to it, as well (or instead).
And that’s absolutely ridiculous. I’m happy that the applications I use most often to do critical work do respect the original function of the insert key; those that don’t (like what I’m using right now to type this) are crippling my ability to rapidly control what I type.
It takes less keypresses to overwrite depending on how you are editing. Imagine how many keypresses it takes to hit backspace, and you’ll understand.
Imho, it’s the same distinction as the difference between backspace and delete.
Back in the DOS days, overwrite was the default text entry mode. Pressing the insert button made the cursor taller.
Today, on windows CMD it is the opposite, the default is insert mode with normal cusror and if you press the button, tall cusor with overwrite mode.
It’s mostly a historical thing, overwrite was more useful in a non-GUI environment. On one keyboard I had at work, I removed the insert key as it was too close to backspace, and I kept hitting it by mistake.
At least the insert key actually works in a lot of text editing apps. Who here has ever used an application with scroll-lock? And why does every keyboard have an LED that comes on to tell me this completely non-functional mode is active?
I use it on stupid forms with underscores as the fill in line. I insert and overwrite the line so it doesn’t make the line get bigger.
Okay, so what are “Pause Break” and “Scroll Lock” for?
break was used to stop software.
scroll lock was used to determine if the arrow keys moved the cursor or moved the contents of the window.
I believe overright was used in some sort of programming (hex code?) also where there was only allowed so many digits per line, you could not enter more without deleting.
When you TYPE a .TXT file that’s too big to fit all on the screen at once, you use the Scroll Lock key so that the text doesn’t scroll past too fast for you to read it. Assuming you’re still using DOS.
On an app I use at work, you can use Shift + Ins to paste from the clipboard, where using Ctrl + V gives you an unwanted consequence. But that’s a workaround for bad software design rather than a good use of the Ins key, IMO.
I’m pretty sure you can un-map the Insert key on most versions of windows so that it does not turn on text overwrite, but I’m not sure how to do this.
Pause Break is still used, for example, it interrupts execution of a VBA macro. Scroll Lock is used by Excel–if you press it, then using the arrow keys will scroll the screen instead of moving the cursor to a different cell. It’s a bitch when you press it by accident and don’t know what’s going on.
On my editor it certainly does toggle between insert and overwrite modes. And I use it, not a lot, but occasionally. Also ctl-ins is used to copy from the clipboard to whatever you are writing to. I use that a lot; it is generally easier than bringing down the edit menu and clicking on paste.
ctl-v for paste is not available? Just curious, on your editor.
Now, for the hard one: What the hell is SysRq (System Request) ever used for? Usually, nothing. Sometimes, magic.
It was an experiment that failed, in fact: It was meant to be used in multitasking OSes (operating systems that allow you to run more than one program at a time) to enable you to switch tasks. (cite) However, since no multitasking OS for the PC has ever actually worked that way, the original use was forgotten and so was the key.
Emacs listens. I 
Emacs.