What ever you said is correct, we can not accelerate neutron with the help of the cyclotrons. Since neutrons have zero charge which means they are neutral particles.

It is may not possible to accelerate neutron, but they can be generated with the high velocity. By using the nuclear reactor, we can generated neutrons with high velocity.  In a nuclear chain reaction, to be more precise, in a nuclear fusion reaction, neutrons are teased with certain energies. From this we can conclude that neutrons can be accelerated but we didn't have perfect method and proof to accelerate.

See the neutron generation here

LINK: https://www.youtube.com/watch?v=1sQX1st5bbw

The process in Cyclotrons:

https://www.youtube.com/watch?v=cNnNM2ZqIsc