I am a deer hunter and an upland hunter, in Minnesota. I only deer hunt north of the rifle/shotgun boundary, in wooded areas. I do have relatives who hunt in the shotgun zone. This is just my personal view, not a claim of what is right or wrong. As with many things, it seems like a subjective topic.
As a deer hunter, I wouldn't want a dog running around in the area I am hunting in the woods, because I think deer are going to avoid that area with the dog scent there. I once witnessed a man get a dog out of his car to join him after he shot a deer in the St. Croix State Park special hunt, as he quarter the deer (yes, quartered a deer) and made several trips back and forth. My daughters and I didn't see many deer from our ground blind, that was a couple hundred yards from his route of travel, after that, for some reason.
As a bird hunter, I wouldn't hunt grouse, in the public woods, during firearm deer season. There are limited parking areas in many places, as well as deer hunters entering from non-designated parking areas. The deer hunters are, for the most part, keeping distance between where they are setting up in the woods, for safety reasons and to have their own area to hunt without disturbances by others. In the public woods, most deer hunters are stationary, most of the time. As a bird hunter, I wouldn't be able to avoid taking us past deer hunters. I might see orange and veer off, but we're still spreading dog scent as we go. Although I do have orange on my dog at all times (when ruffed grouse hunting and pheasant hunting), I could see a bored, unethical hunter, thinking the dog is by itself and taking a shot at her. That should be a very small minority of deer hunters who might act that way, but still a risk to consider. Without orange on the dog, then there is the higher risk that they shoot at the dog, thinking it is something else.
Now, switching out of the woods, to the southerly fields, it is a bit different. I have already come across bow hunters on public land, both in South Dakota and Minnesota, this year, as I have in past years. Just like if I see a vehicle at the public land parking area where the owner is most likely a bird hunter, I am not going to park there and hunt. That's for safety reasons and sportsmanship reasons. I'm less worried about a bow hunter taking a pot shot at the dog, out in the open field, where they will most likely see us ahead of time. So it isn't about risk to the dog. Once we add shotgun and modern muzzleloaders to the equation, with rifled barrels, the risk to the dog and to other hunters increases (usually not many trees to stop projectiles like there are in the woods).
However, if it is a big area, I might go to a different access, like I would if it was a bird hunter parked there, if I think the chances of us interacting would be small. In other words, if there is so much land between one parking area and the other, it would be unlikely for me to hike far enough or the other party to hike far enough to run into each other. Likewise, there might be water separating the two sections of the same property.
As a deer hunter, if I was hunting in open country, I would actually appreciate someone bird hunting on nearby properties (let's say the deer hunter is on private land, and the bird hunting is on public land), or bird hunting on the other side of the same property. Deer are often lying down during the day. Bird hunters could get the deer up and moving for the deer hunters. I am not suggesting a coordinated hunt, using dogs to aid the deer hunter. I'm saying if you are independently hunting, it has that added benefit for the deer hunters. I feel the same way when hunting in the woods, when it comes to people moving around. As long as they make an attempt to veer away from my stand, having people walking around in the woods can get the deer up and moving.
Bird hunters can often be a good source of information for the deer hunters too. Most years, whether it is a firearms hunter or a bow hunter, I end up talking to at least one deer hunter when bird hunting. They will invariably ask if I've seen any deer, and I will share what I've seen, and where.
All this is to say, personally, if I see a deer hunter parked at a public pheasant hunting spot, I am not going to park next to their vehicle and hunt that area of the property. I wouldn't do it if I knew it was a bird hunter either, and my behavior doesn't change based on what they are hunting (might be a waterfowl hunter, too).
From a strategy perspective, I might come back to the property later and see if they have left. I'd probably stay away from wooded areas on the property, in case they are coming back in the evening or following mornings.
If the question is whether to stay off all public fields during deer season, because deer hunters may want to start hunting a spot I am hunting after 9 AM (10 for SD, 8 for Iowa, etc), my personal opinion is: it is a shared resource. They can find another location, just like I would find another location if they were there first. However, the nice thing is most deer hunters are going to beat the pheasant hunters to the spot, so we have a built in way to avoid interfering with each others' hunts by merely not hunting where someone else is hunting, regardless of what they are hunting.