Many people do things pretty similar. They may house their chickens in the garden during the winter, have a tractor that fits over the raised beds, or have the run divided into sections and use a different section for the garden each year. There are always lots of different ways to do things.
I don’t know where you are in Texas or how moist or cold your winters are. That stuff needs some moisture to break down into compost. If it stays frozen it won’t break down very well. But people a whole lot further north of you empty their coop in the garden in the fall and it is still ready to plant in the spring. It probably does most of the breaking down after the ground thaws and maybe even some after it is planted. I’d think in Texas this would not be a problem at all as long as you get some moisture.
Pure compost is not a great soil for a garden. It does not have the structural strength to support the roots so things can blow over pretty easily. It can also dry out pretty easily. Compost helps sand and clay soils manage moisture better but it’s better mixed than by itself. Your chickens and turkeys will probably take care of that for you by scratching.
Another thing is that that stuff really shrinks in size when it composts. What looks to be a huge thick layer of compostable materials winds up making a fairly small volume of compost. Even if you pile the leaves and such in there pretty thickly you won’t wind up with a hugely thick layer of compost so they should easily mix dirt in with it for you. I’ll just mention this so you are aware of it since I’m not looking at what you are doing, but I don’t think it will be a problem for you at all.
Now the poop. Chicken and turkey manure is fairly hot with nitrogen. If it is too fresh and too thick it can burn plants and even kill them. How hot it is will depend on how well it is mixed with carbons like those leaves, how long it has been there, and how much moisture there is in the mix so the microbes can do their job. It’s not the kind of thing you can just mention a time period and it will be OK because there are so many variables, such as how many birds pooping over how big an area and how well it is mixed in. So you will need to remove the birds a bit before you plant it so the poop can break down some. How long? I don’t know your details but I’d think a couple of weeks would be enough to not stunt or kill the plants. If it’s just a few birds and not a tiny area, you might need almost no time.
Is it safe to eat the produce or save the seeds? Oh my yes. What’s the quickest thing you are going to harvest? Radishes or maybe lettuce? Those will take a minimum of three weeks from planting to harvest. By that time any fresh poop should have broken down so it’s not a possible contaminant, especially if you let it sit a week or two before you plant it. Most poultry pathogens won’t affect humans anyway, pathogens are normally species specific, but some can. Any garden soil, whether it has compost or fairly fresh manure or not, can possibly contain pathogens. It often contains some fresh poop too. Birds and various creepy crawlies are pooping in there all the time. It’s always a good idea to wash the produce before you eat it. I use a lot of compost that I make and every few years I empty the coop on the garden in the fall. It doesn’t bother me to pull a fresh radish or carrot, wipe most of the dirt off on my dirty jeans, and eat it. I grew up doing that so I have a strong immune system but it is always safer to wash it first. As long as you wash it you should not have a problem.
One potential problem. Your birds will scratch in there a lot. If your rock wall is not high enough or the mesh on any wire fencing is not tight enough, they can scatter that composting stuff all over the place. You can wind up with that stuff scattered all over the place, not necessarily in the garden.
What you are talking about sounds great. Go for it.