Finally, a comm for that one user who hand-makes longbows. This ones for you, comrade.
The light came with an alternate mount that screws on to the handlebar but it makes taking the light off more annoying, and obviously removes the ability to freely put the light on different bikes
I was hoping this design was universal but none of the straps I'm seeing online have the screw hole. Well, there is one very similar strap but I'd have to pay 13€ postage on a 5€ item which offends me on a spiritual level
Speaking from experience here; your problem isn't gonna be the strap, it that it fits to the screw interface on all points. While they all look 99% similar, they're not, unless this is a Lezyne or something.
I'd advise you to buy a generic "flashlight bicycle mount" from wherever, something like this;
Has the added bonus points for if it holds up and your light doesn't you can also just use it for the next one.
Alternatively glue and ziptie a garmin quarter lock mount (knockoff, obviously, but using that standard) to it and get yourself a female receiver to attach to your bikes handlebars. They're all ubiqioutous and therefore well cheap.
I think it might be a Lezyne, it has almost the same design as mine but slightly longer and thinner.
But yeah, those flashlight mounts will easily do the job, and you can find them super cheap.
It's not a Lezyne (searching for the brand of the light itself I only get Finnish/Nordic results so it's probably a design licensed off to a local company), but the very similar looking replacements I found were Lezyne ones. The only problem is that the postage fees for the straps are stupidly high in proportion to the actual price of the item. I'd pay 5 euros for one but not 5 + 13 euros. That's just nuts when I can get shit mailed from China for 3€ or free
(searching for the brand of the light itself I only get Finnish/Nordic results so it's probably a design licensed off to a local company)
They probably just copied it, there is only so many ways you to skin a cat and all.
I just wish someone in China had copied it and put it on Ali 
To me - that looks like something from a sci-fi movie about the size of a smallish PC tower and it's called the "device".
I'd advise you to buy a generic "flashlight bicycle mount" from wherever, something like this;
You mean ones like these?
https://www.aliexpress.com/item/1005007321929071.html
https://www.aliexpress.com/item/1005005843902557.html
The only problem I see is that flashlights are generally thin and round whereas my headlight is chonky and square. I'm not sure these would fit around its girth.
Alternatively glue and ziptie a garmin quarter lock mount (knockoff, obviously, but using that standard) to it and get yourself a female receiver to attach to your bikes handlebars. They're all ubiqioutous and therefore well cheap.
I seem to have trouble finding the male bits on AE while female ones are easy to find, gonna search some more
You mean ones like these?
The only problem I see is that flashlights are generally thin and round whereas my headlight is chonky and square. I'm not sure these would fit around its girth.
Yeah. The chonk should help to keep it in place, honestly. Unless the strap breaks, in which case on chosen examples any old rubber band would do.
I seem to have trouble finding the male bits on AE while female ones are easy to find, gonna search some more
https://de.aliexpress.com/item/1005007446027044.html Here ya go
Thanks! Both of these solutions are cheap enough that I'll probably try both of them out
You might be able to staple it. Super glue might also work.
Do you know someone with a 3d printer? A replacement could be produced with a flexible filament.
Voile straps come in a lot of different sizes, and the holes will work in your favor with the screw mount.
These things are amazing. I keep a few dozen of them in my car, and they've saved my ass a number of times. Once, they held the suspension together on a 12k lb work trailer until it could make it to a garage.
You think these knockoffs would do the job?
Since it's a bike light and not something you'd die without if it fell, I'm sure the knockoff would be fine, but I've only ever used the original ones from Voile. I've never had one of those break on me.
Reusable zip tie
You can find almost anything bike related on aliexpress, If your search words aren't working, you can take a picture of the item and do an image search on aliexpress
In this specific instance AE let me down 
Could possibly order another from Amazon and then return it with the broken strap, keeping the new one maybe. If you don't need that1st hole (the broken one) I'd just tape it as Hestia recommended. Could maybe tape the whole thing over and then cut out the hole you need, even.
I don't see how tape would be an improvement over the hard plastic mount.
I'm looking for a solution that allows me to attach and take off the light quickly and put it on other bikes when needed
No, tape the strap itself back together and then continue using it as before. The more and tighter you tape it the less elastic it'd be, but maybe you can wrap the middle of it with tape and sacrifice that hole. If that's possible, hard to know how do-able that is without doing it myself.
Duct tape
The screw thing is weird. Does the screw head just sit straight on the handlebar, scratching it up? Is the front bit rigid, providing a platform, or is it just basically all silicone?
I agree with the idea of getting a garmin quarter lock mount. It's a handy way to attach things, lots of cheap generic options, including those using silicone rubber bands, and it makes the challenge as simple as drilling a hole, assuming the mount doesn't have the right size for that screw. You can get this on Aliexpress for a couple dollars with either free shipping if you buy a few more dollars worth of stuff or a couple dollars shipping.
The screw is for attaching the rubber mounting strap to the headlight itself. The light came with two different mounting options, the strap and a long plastic mounting arm with a Garmin mount on top and a separate mount for the headlight below. There's an alternate hard plastic bit you can screw to the bottom of the headlight for that purpose. You then use a thumbscrew to attach the light to the mounting arm
It's just much less versatile than the strap was