Would the British have won the battle of Saratoga if Gen. Howe hadn't foolishly sent his army into Philadelphia instead and adhered to Gen. Burgoyne's strategy of linking up with his army coming down from Canada? If the British had won the battle do they win the war as well?
Not necessarily. Remember that if Howe goes up the Hudson, then Washington follows him, so you've got a "super Saratoga" of Howe plus Burgoyne vs Washington plus Gates.
The British problem, though, if that even if they win, the Upper Hudson area lacks the supplies to provision such a large army, which presumably has to return to New York. In that event, all or most of the Hudson valley is likely to return to American control. If Howe leaves garrisons in Albany etc, they risk being picked off one at a time once the main British army moves away. However crushingly the Amercans are defeated, at least some of their units are likely to reform somewhere or other, so it ain't over.
This was the fundamental reason why Britain lost the ARW, despite winning the vast majority of the battles. Just too much America and not enough British army. However well we did, we could never occupy more than a corner of it at a time.