If I understand correctly a fertilized egg will not begin to develop until it begins to be incubated. Therfore keeping them on the counter will not reveal any developement after, say, a week when you go to make an omelet.
Pretty much. The warmer the egg is kept, the faster it develops. How warm is your counter? It'a possible if the temperature is up in the 80's you can get some development, but it will be slow.
With this in mind, provided that is correct, if suddenly you had a broody on your hands how long could fertilized eggs be setting on the counter and then set under a potential mama hen?
There are certain ideal conditions for storing fertile eggs. The closer you are to the ideal the longer the egg can be stored before incubation starts. A hen can hide a nest on the ground and lay eggs for a couple of weeks before she goes broody and still hatch most of the eggs under certain conditions. The ground helps moderate temperature and humidity. In some conditions it will not work as well even for a hen. There are no absolute with any of this.
According to the experts at Texas A&M an egg does not need to be turned the first few days of storage but after several days it helps. A hen hiding a nest to lay enough eggs to start incubating is not turning them several times a day yet the experts say we should. I'm not a broody hen so what works for them does not work for me. Still turning them is considered part of the ideal conditions for humans.
I typically store eggs for hatching at room temperature and humidity (pretty dry most of the time with heat or AC) for up to a week before starting them, temperatures in the 70's. I do turn them two or three times a day. I usually get really good hatches. Since I'm not that close to ideal conditions my hatchability drops if I go much longer. That does not mean that none will hatch, just that fewer hatch. When I have a hen go broody, I then start saving eggs for her to hatch instead of taking them off my counter storage. Part of my test is that a hen has to stay on the nest for at least two consecutive nights before I deem her worthy of hatching eggs. That gives me tie to start collecting eggs specifically for hatching. If my outside temperatures are pretty hot or cold, I collect a few times a day to get them out of those extreme temperatures.
When I was a kid Mom stored the eggs in a bucket under the kitchen sink. When we had a broody hen that Dad wanted to hatch I'd grab a dozen eggs off the top of that bucket, mark them, and give them to the hen. No special storage in any orientation and no turning. We got great hatches but those eggs were no more than a couple of days old.
Which brings up my next question. Can I tell if an egg is fertile before begining incubation without cracking it open?
There is supposedly a laboratory in Canada that can take a sample of the inside of the egg and not only tell you if it is fertile but what sex it is by DNA analysis. I don't find that very practical. The only two reasonable ways I know of to tell if an egg is fertile is to either crack it an look for the bull's eye or to incubate it and see if it develops. Candling an unincubated egg will not work.