Could time illusion be used to send a spacecraft faster than the speed of light?
Could the time illusion be used to send a space [probe] slower than light but have it seem like it is traveling faster than light?
Yes, but it is of no use whatsoever. [**]
There's nothing special in the behavior of this particle jet that couldn't be done by a space probe. To make this happen we need to send a space probe to a point far out, turn it around and make it race towards us.
The effect described is nothing else than a variation of the well known Doppler effect. The sound of a vehicle speeding towards you sounds higher than the sound its horn can produce. In the same way, the image of the space probe seems to approach you faster than the speed the probe can reach:
Assume a probe is flying towards you. It travels for 100 years covering 99 ly at 99% the speed of light: Light emitted at the end of the journey reaches you "now". And light emitted 100 years earlier at the beginning needed 99 years to reach you, i.e. arrived 1 year ago. So, the image traveled 99 ly in 1 year, 99 times the speed of light.
Also note that the effect strongly depends on the direction the probe is traveling in - an observer at the starting point of the journey sees the image moving away from them at only 0.5 times the speed of light. Light emitted at the end of the trip arrives at this observer 99 + 100 = 199 years after the start.
In conclusion, you can use this "time illusion" to make the image of a probe move faster than it actually does. But it all breaks down as soon as you take not only distance but also the time light needs to reach you into account properly.
[**] Apart from science fiction, where a war-faring, sub-luminous space civilization using long distance weapons would profit from it: The enemy wouldn't be able to detect their 99.9% c projectiles in time but only see them racing towards them at 1000x the speed of light in the very last moment.