Generally speaking, would replacing the "arrow" for the balance cock make the watch run again?

Edit: Arrow -> Regulator Index

 

the regulator can be replaced of course. This does not effect the general "ability" of working of the movement, its just for regulating...
does the balance move freely?

 

Simple answer: No.

 

Thanks for the replies. The balance moves but the regulator index "ring" is loose and doesn't stay in place. I'm also noticing the watch runs when the crown is pulled as well..

 

Your watchmaker might be able to sort it out for you.