This appears to be an artifact of the way the script changed between the draft and the final product. In the original, the Sphere was merely one of a fleet of Borg ships that invaded the Sol System. During the battle this spherical warship shrugs off several shots, then heads off toward Earth, ploughs through the orbital defence systems and then does a time-jump as it enters the atmosphere. It seems that this was their secondary intention all along, assuming the fleet failed to achieve its primary objective by assimilating the Earth in realtime.
In the film, however, a single menacing cube arrives (like it did in TNG) and the sphere is contained within the cube as an escape craft with limited defences. The Queen initiates a time-jump as a last-ditch measure to take her out of the firefight. give her an unopposed shot at assimilating the Earth in the past. Not an ideal scenario, but acceptable.
This gives us two distinct reasons why Picard didn't destroy the Sphere immediately;
In the original script, they didn't challenge the Sphere because it wasn't in the immediate vicinity of the fight, having gone off in another direction shortly after engaging.
ADMIRAL HAYES: The new quantum torpedoes are doing the trick, Jean-Luc. We've destroyed forty-seven Borg ships so far... and only lost fifteen of our own.
But one of the Borg ships has broken through our defenses, and it's heading directly for Earth. Can you handle it?
In the revised script (on which the film's novelisation is based), the Sphere remains unchallenged because it was mistaken for debris, then hurtled past the Starfleet vessels before they had much time to react.
Oddly, the sight brought little comfort—a mystifying reaction until he
saw, emerging from the flying dust and shrapnel, a smaller vessel. Not
a cube but a sphere, with the same Borg disregard for aesthetics as
its predecessor, the same peculiarly unappealing leaden color, the
same honeycomb design with exposed circuitry and tubing.
The sphere flew past the assembled starships directly toward Earth.