will all these jakes shut the big toms down?
That's all part of the progression as turkey populations grow and last year's hatch parlays into this year's social structure. In Missouri, where they have 650,000 birds, I've watched large packs of Jakes for years. The gobblers won't leave the county; as Ray Eye says- "They're just out there bein' turkeys." The MO birds don't gobble much after leaving the roost anyway, there's so many hens they don't need to. Excess gobbling just brings predators or a butt kickin'.
What you end up with is a lot of silent gobblers who are still there, and will still strut and come to the call, but won't advertise their presence much. Hunt the mid morning strut zones and haunts where the hens feed and you'll still find gobblers.
I would think QDM (Quality Deer Management) works much the same way. If you get the buck to doe ratio close to 1:1, you stand a real good chance of calling an older buck in. But if there's a ton of does and lot's of younger bucks, your odds go down dramatically. Likewise, if there's an abundance of hens and packs of jakes everywhere, the tom's need to gobble goes down proportionately.
The difference hunting here in Michigan, where we only have 1/3 as many birds, always amazes me. Gobblers will answer your call almost any time of day. Are too many turkeys a bad thing? Possibly. It sure takes some of the fun out of it when they won't gobble. Let us know how your season progresses.