There is a series on the American War of Independence currently showing on BBC2, presented by Professor Richard Holmes (lecturer at RMA Sandhurst). It is balanced, and very clearly presented, with lots of action shot of re-enactments.

Holmes maintains that Saratoga was a strategic disaster caused by lack of proper coordination between British commanders.

Some additional reading I have done sugggests that the fault lay with The Secretary of State for War, Lord George Germain, who as Lord George Sackville, earlier acquired the title, "The coward of Minden".

At any rate, Burgoyne proposed to Germain that his army attack south from Canada, while (Sir William) Howe attacked north from New York, and they would meet at Albany. Meanwhile, Howe proposed that Burgoyne stay put, while Howe and Cornwallis went south to attack Philadelphia. Germain agreed with both of them, then went off to his estates. There was no chance to issue clarifications or corrections before both campaigns were launched.

So, Burgoyne attacked, but there was only a small detachment under Sir Henry Clinton left in New York, which could not make much more than a demonstration to help Burgoyne. Burgoyne seems to have been at fault in pressing forwards in spite of obvious indications that all was not well (no loyalists joined him, supporting columns and detachments were routed or foiled, he could not collect supplies from the countryside).

Given the numbers that Gates's army eventually reached, and their effectiveness (especially of Daniel Morgan's riflemen), I doubt whether Burgoyne's plan could ever have worked; though the result would have been merely a repulse, rather than the surrender of Burgoyne's entire army.