The problem is 7 of 9's statement in the episodes YOH and Relativity. Where she says the Borg were present during the events of FC.
The collective would have been in contact with the sphere and known that it was going to travel back in time to stop first contact.
Remember the plan was to assimilate Earth in the present. Using the sphere and time travel was IMO something the Borg came up with on the fly, after their massive cube was destroyed.
The big plot hole of FC is that the Borg could've stayed in the DQ and used a time sphere to travel back in time there and THEN assimilated Earth in the past. It's a solid plan, and from FC we see that it can work. Why the Borg never tried to do it again all throughout VOY is beyond me.