Why can't it unfold etc in Earth orbit, where a repair mission can be sent if needed, and then travel to its Lagrange point?
To get from low-earth orbit to the sun-earth Lagrange point 2 (where the JWT is headed) takes around 7 km/s of delta-v[0]. That's a lot of speed.
You could try to do this gently enough that the unfurled JWT won't be damaged by the acceleration. This isn't totally impossible, but you'd need a good Hall thruster (ion engine) with a huge amount of reaction mass, since the JWT is so big itself. It would need to run for longer than any other such thruster has. It would need massive solar panels to power it.[1]
Or you could have the original rocket just be bigger, and throw it all the way to the right orbit while everything is packed tight.
[0] https://en.wikipedia.org/wiki/Delta-v_budget [1] I'm guessing at this, but that's my intuition. I encourage anyone to correct me because space is too cool to be upset that I was wrong.
Does that really matter without air resistance?
Depending on how high you actually bring it. Like 500km away from earth is still an orbit (I think that's Hubble's orbit) but how much force do you need or will happen?
Otherwise it would just take longer.
And to answer your question in the other response, many thrusters have a minimum thrust, and even that minimum may be too much for the parts when deployed.
As an extreme example, imagine a stick one lightyear in length: if we ignite a rocket on one end, firing perpendicular to the stick's length, then the other end cannot start moving for at least a year.
The less Gs you need to design a component for, the lighter/simpler it can be, so why unfurl early and add that extra mass and complexity to the design?
A broken JWT could wait a few years in orbit.
Looks like a reasonable repair cost (and only if) it turns out to be broken.
So what you're saying is, this could easily be funded by some billionaire, e.g. Jeff Bezos who already sells billions of dollars in Amazon stock per year to fund Blue Origin?
Not saying this should be done privately, but if funding is the problem, that problem can be solved.
Space travel is less expensive than most people think, it just isn't very high up on our list of priorities.
Another comment mentioned that it's not designed to accelerate while it's fully deployed, and that's true enough. You'd wreck it.
The other essential thing is that there's no way to give it and its instruments anything like their designed operating parameters (pretty hot on one side of the sunshade, something like 40 kelvin on the other side) in Earth orbit.
Let alone the current lack of in orbit service capabilities like we had when the space shuttle was still around.
I saw some interviews of engineers of the jwst and few of them had similar ideas, or at least to assemble them in leo then slingshot them to their final positions/orbits
Another booster up in the fairing? That would need to be quite heavy. It would be a totally custom thing for this specific mission. You would need a suitable storable propellant.
Leave the booster from the Ariane attached? The lh2 and lox would boil off after a few days.
What do you do if the deployment fails? SpaceX's dragon can't do space-walks on its own, it doesn't have an airlock. There would be no way to fix it short of developing a whole new space craft for that task.
Not that Armageddon was particularly focussed on reality.