The problem was that Edward did make a strong effort to influence the British foreign policy, and it had an effect both at home and abroad. IMO Clark makes a compelling case at Sleepwalkers: Edward had larger impact to British foreign policy than Nicholas II had to Russian foreign policy.
Do you have an example of Edward's impact on policy? A meeting where he directed the PM of the time to seek better relations with France?
He might have had an impact, but it was more of a
confirmation of British policy as opposed to
direction. British traditional strategic interests dictated they seek an alliance with the Franco-Russian bloc (no one power should dominate the continent, British power should ally with the weaker faction) Edward's francophilia merely supplimented the policy of Lansdowne.
It would be interesting to play out a TL that sees a British monarch opposed to the policy of the government, but I would guess - butterflies notwithstanding - it wouldn't lead to much of a change.