Welcome!
I use accelerator all of the time for CA finishes, but as you see, others do not.
I use the aerosol can for application, but I have never tried the pump variety, so my recommendation needs to be taken with a grain of salt.
I had problems at first that I thought could be tied to use of the accelerator, but decided it was over application, not the accelerator itself.
My procedure is to apply the accelerator while the lathe is turning slowly, and only use 2 very quick shots.......emphasis on VERY QUICK. I then feel the blank with my bare finger as it still spins and don't apply the next coat of CA until I am sure the last one is dry. You will feel a slight drag on your finger if it isn't dry.
I also have my lathe set up with positive dust collection, so I am not bothered by any of the chemical issues of either the CA or the accelerator.
As I said, there are lots of opinions out there. This is just mine.