Why are baking measurements such nice round numbers?

Realistically, there's a lot of flexibility in baking, in spite of the cargo cult mantra that you have to follow pastry recipes exactly. There are simple, weight-based ratios that can be used as a foundation for plenty of variation. The "round numbers" are, in fact, approximations, and this is often why you'll see that a typical baked good recipe in a consumer-oriented cookbook isn't designed to scale exactly past, say, doubling or so.

When you diverge dramatically from a base ratio, you may get an unexpected outcome. But, for example, pate a choux can be converted from the basic cream puff shell to a gougere by simply adding a modest amount of cheese to the dough. Within a moderate range, the exact amount of cheese won't matter that much. When I make muffins, I follow a basic ratio, and add additional items like fruits and nuts without even measuring; given enough experience, you can eyeball how much a recipe can take.

You can see more evidence of this as you look at most cake or pastry cookbooks. If you reduce the components to their essential character (fat, flour, egg, sweetener, liquid, leavening, flavoring) you'll probably find that there are only about 6-12 archetypal recipes in most such cookbooks. Some of them are even up-front about it and tell you to start with the "basic yellow cake recipe" then add x, y, and z.

That's not to say you won't get different results when you use less or more of an ingredient than the base recipe requires, but you'll probably get very adequate results as long as you're close to the base ratio.

Most chefs aren't particularly scientific, and you should realize that many of our archetypal baking recipes are the serendipitous result of relatively haphazard experimentation. Supposedly, the original souffle was a "mistake" caused by using too many eggs in a cake recipe. Additionally, even with weight based measurements, most ingredients have quite a lot of natural variation, depending on seasonality and, even for things like refined flour, climate and varietal differences can change your results. I've started with some recipes that worked pretty well when I was a student in Germany and had very different (though often acceptable) results in the US with similar ingredients.

Ruhlman's book called Ratio is a good place to look at these kind of archetypal recipes, and includes a fair amount of references on baked goods, though it's not the focus of the book.

I've always been taught that baking is a science when it is compared to cooking. Cooking is very much 'to taste' and very individual. There are not as many things that can go wrong with a standard recipe in cooking, and you have a lot more room for creativity. You don't have to look at baking as that precise. However, unlike cooking, where you can add or subtract from a recipe with no real harm to physical structure, that does not hold in baking. For examlple, if you feel that there is too much salt in a recipe, cutting back can (and most likely will) have a cascading effect through the ingredient chain. You have to understand your ingredients and the effect they have on other ingredients. That is what makes baking a more precise science.
And in terms of flour, it is often the most 'ranged' ingredient. Depending on flour type, miller, altitude, water temp. etc., the amount of flour in a given recipe is always a guide. Again, you have to know your ingredients. You will never see 'one and an eight cup plus 2 TBSP hard flour' in a recipe, because it is so variable for many different reasons. Flour and water are the two most flexible ingredients, and are always variable. Hopefully this helps, I am sure someone will come along with a more scientific explanation for all different ingredients, I am just offering an experience based answer to your question. Also, you are right, do not mess around with baking soda and powder. If you do, you are asking for trouble. I would suppose that the rule of thumb would be that the more exacting an ingredient is (tsp, quarter tsp), you don't want to change much.

Of course there is some wiggle room.

Depending on the type of recipe you are baking different ingredients are going to be more important. In bread, for example, the flour to water ratio is going to be the most important. Even slight changes in these can dramatically affect the product. In my experience, for my 6 loaf recipe, a "slight" change would be more than 1/4 of flour or just a couple Tbs of water.

Keep in mind that haphazard measuring of flour can result in differences far exceeding 1/4 cup. (depending on the size of the batch of course)

Some ingredients aren't so picky- the difference in egg sizes, for example, has never significantly impacted my results; yeast quantities affect rise times, flavor, etc. but it will work out; Salt is to taste.

I bought a digital scale and converted my old bread recipe to weight and now I don't have to measure anything (the old way) I just dump in the flour until the weight is right. It's nice.

All in all- if you are measuring so carefully that it isn't fun anymore then you're taking it too seriously. Measure carefully when it's important but let it be enjoyable.

Basics: Baking recipes are FORMULAS i.e. total flour weight = 100% all other ingredients are a % of the flour weight i.e. 80%, 120% etc. this allows you to scale up/down "recipes" then the intangibles of experience 'taste, feel, smell, etc.' allow minor adjustments.

The flour is scaled at 100% because of all the variables affecting the water content of your flour i.e. humidity, type of flour, grind and others.

See: wikipedia.org/wiki/Baker_percentage