JWST isn't a probe like Voyager. It's job isn't to go to other worlds and see what's there. Its a space observatory. It does exactly the same job a telescope on earth would do, just in space. The reason why is to avoid light pollution and the interference from earth's atmosphere.
So that fuel isn't used for going places, its for aiming the telescope and keeping it steady. Once the fuel is gone it can no longer control where its pointing. If you cant aim a telescope then its useless.
You can aim it with reaction gyros which rotate the telescope and require only electricity. What you cannot do with electricity is translate the telescope along a vector to compensate for orbital decay and keep it on station. That's what fuel is for.
I understand it's not designed for exploration, but my thinking is given the other alternative is burn it up in the atmosphere isn't it better to get a bit longer (few years?) use as it drifts off vs burnt up.
With the you need to aim it, could they sent it to a slow spin/wobble and time image capture as focus would occasionally pass by things of interest? Not sure if it needs to be relatively stable for that long timed exposure.
There is probably good reason for not doing this and trying to understand why.
Don't worry, JWST isn't going to burn up for a few billion years. Until the Sun expands into a red giant at least.
They are sending it to the L2 lagrange point which is further out that the moon. That's also why they are so careful about making sure the origami sun shield works right the first time. If it screws up the whole mission is lost because there's no chance to fly some astronauts out there to fix it like we did with Hubble.
As for exposures while its spinning, I was under the impression these are really long exposures and require it to be precisely aimed for quite awhile.
> why they are so careful about making sure the origami sun shield works right
They cannot really test it on Earth, as it must work in free-fall. Given the past problems with unfolding these things, I have a horrible suspicion that it won't work, and we won't be able get to L2 to fix it.
So that fuel isn't used for going places, its for aiming the telescope and keeping it steady. Once the fuel is gone it can no longer control where its pointing. If you cant aim a telescope then its useless.