It's a problem of energy and orbits.

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.