Actually, most veterinarians, including dermatologists, will tell you the exact opposite - that testing for food allergies using intradermal or serum allergy tests are pretty much not accepted in todays veterinary medicine (though there is at least one poster on this board, who to my knowledge does not have a medical degree, that disagrees with me) - and using these tests are WIDELY accepted by both general practitioners as well as board certified veterinary dermatologists to diagnose environmental allergies.
There are a few labs that are known for their reliable results - I choose to use HESKA, personally - I have had very good results with them and they are reliable, I've had good feedback from boarded derms, and they have good customer service and derms on hand via the company for me to go over cases with (including my own dog).
FOod allergies are grossly overdiagnosed by the general pet owning public, and they rarely have gross SEVERE dermatologic signs like you mention here - *most* truly food allergic dogs do not come across as severely pruritic (itchy) all over - they tend to have a few spots they are itchy (including around the anus, and sometimes chronic ear issues), but by far the most common allergies are atopy - environmental ones. Many food allergy dogs can have GI signs.
Typically I start all dogs off, for me, with a skin scrape in multiple places - I may even treat them with revolution (since I can always be wrong with a scrape and selamectin really didn't hurt most animals and can only help as well as ensuring adequate flea control for a possible flea allergy!), and skin cytology - if I see bacteria, I will give antibiotics, if I see yeast I will treat topically with antifungals or potentially orally if bad. I will check ears to see if I need to address otitis. Check between food pads. Check distribution of the itching. Discuss diet, seasonality. Oftentimes because it is easier some people will opt for a full food trial before working up an atopic animal because it's always easier to rule that out then go to it later. Once I have ruled out a food allergy, there are a multitude of ways to deal with an environmental allergy.
REALLY, I do stress allergy testing - even if you don't go for allergy shots, it's nice to know what the animal is allergic to because environmental control can be key. Some things can be done to modify the house/environment (I got AC to help my dog, I swear!) including HEPA filters, pulling up carpets, getting rid of featherbeds, pillows, etc, changing the dogs beds or where the dog sleeps, altering where you walk the dog, etc.
Allergy injections do become an option - typically in my experiences 50-75% of animals will experience 50-75% resolution of their symptoms. You never cure an animal of allergies, you simply manage them - so you can hope to decrease the severity or decrease the number of outbreaks with the injections, and manage them medically with courses of antibiotics and topical shampoos, antihistamines, and potential steroids if needed intermittently
Another option is Atopica (cyclosporine) - expensive, with less side effects than steroids, but works very well. It modifies the bodies response to the allergens - it's what we use in human transplant patients to prevent rejection.
Take this from me - I'm not just the president, I'm also a client
I mean, I'm not just a vet, but I'm the owner of one very itchy allergy dog. My own dobe has severe environmental allergies. Like I said - true allergies are not eliminated, they are managed - you will likely always have outbreaks, we just try to minimize them and make them easier to deal with. I'd really urge you get your puppy off prednisone and to a board certified dermatologist for some real answers as to what is going on - clearly you're not actually getting any answers right now on this path.