The starting gravity and ending gravity serve many purposes, but ultimately will only tell you one thing, the percentage of alcohol. Some of the purposes it may serve are:
- Beer style guidelines
- Mouthfeel, flavor, bitterness, even aroma (FG)
- Yeast tolerance (SG)
- Efficiency of sugar extraction in all-grain brewing (pre-boil gravity)
- Hop efficiency
- How much grain/extract to purchase.
It really is one of the most important measurements of the beer (others include IBU's, and SRM).
If you formulate a recipe, quite often you may wish to create a style. For example, you may wish to brew a farmhouse ale, specifically a saison, you can see that it requires an original gravity between the ranges of 1.048 - 1.065, and a FG between 1.002 - 1.012. In order for your beer to technically be considered a saison by the BJCP's standards, you would need to land your measurements between those ranges. If you want to pull a Dogfish Head and brew whatever you want, toss out any guidelines, and brew what you want.
If you do go with the "screw it, I brew what I want" approach, it still serves other purposes. Your yeast will only be able to tolerate so much alcohol before it becomes toxic for the yeast, and it cannot tolerate it. For instance, you can't brew a Russian Imperial Stout with a low tolerance yeast, because of the tremendous starting gravity of the style. The OG will help you determine what yeast you should choose. If you go too low, the yeast will try to do its magic, but may die out too early and final gravity will be much higher than you want, making it unbearably sweet to drink.
When you all-grain brew, you "mash" the grains, which is soaking them in warm water to extract sugars from the husks. Each grain has a maximum efficiency, and while you'll never be able to reach 100% max, it's nice to not have to compensate with more grain because you had low efficiency. The way you measure the efficiency of your mashing process, is by taking a gravity reading (which will ultimately become your original gravity upon completion).
So as you can see, the starting gravity, and ending gravity is probably the most important measurement in brewing great beer.
As an aside, you shouldn't interpret William Moore's use of approximation as "winging it". While I don't own a copy of the book, I assume he's using approximate readings in one of two ways (or maybe both).
EDIT: Now that I understand what he is talking about, what he's explaining is essentially going to help you determine how much fermentables you need to use to essentially achieve a certain ABV (long-term goal). Essentially you'd be stone-age brewing if you didn't bother with gravity readings. What you'd get is what you get. What he's describing is one of the most basic steps in formulating a recipe.
With that said, while having knowledge of how to calculate your original gravity is very valuable and a great way to impress and confuse your assistant brewers and friends, there are plenty of tools that will do the hard work for you, freeing you up to get back to doing the fun part of homebrewing.
