Trying to serve the code sitewide would wreck the targeting if you want to show widget ads to widget seekers, different product ads for different product seekers, etc.
Users that are to be shown your widget ads get cookied for that when they visit your widget pages, not some other page. Ditto for the other products. If you just use the same code sitewide, AdWords wouldn't know how to target which loses the whole point.
Side comment, but related: Remarketing with followup ads is most effective (and most cost-effective) if you keep a light hand about it. Set tight daily frequency limits to make sure you don't overdo it. User respond best when the effect you achieve is an occasional courteous reminder, not "stalk the user".