The first number with the "W" indicates the viscosity at temperatures far below 0C; to measure the "thickness" it's at -20C for a 15W, -25C for a 5W, -30C for a 0W. To measure the "suckability" the test is at 5C lower than the "thickness" test.
Frankly, the "thickness" portion doesn't matter anymore. When cars were carburetted, they had to crank at a certain speed for the carburetor and choke to see enough air speed to deliver enough fuel to start when it was cold. Therefore the oil had to allow enough cranking RPM. Fuel injection will deliver enough gasoline no matter what cranking RPM.
The "suckability" is more important than before. This is how easily the cold oil can be sucked through the oil pump pickup tube. Like a milkshake, it can be hard to "****" but even worse is to **** a hole in the shake and ingest air. In an engine, that would be bad. With a fuel injected engine that starts at almost zero RPM, if it really does start, but the oil pump can't **** any oil into it... that would be bad!
In practical terms, a low "W" number would be an easier oil to push around, and for the racers looking for the last horsepower, the lower the better. It does not mean you are giving up any high temperature protection.
The second number is the "thickness" at 100C. This bench measurement was determined long ago, and is still a reasonably good indicator now. The better measurement is called "high temperature high shear" viscosity, which is how the bearings would "see" the oil (under high shear and 150C.) However, the current system makes the kinematic viscosity at 100C and HTHS viscosity fall into the same SAE oil viscosity bracket, so you can't really make a mistake anyway.
The more viscosity at 100C, the thicker the oil film. A film that is "too" thick loses fractions of a percent in fuel economy. An oil that is "too" thin allows metal contact, which after the additive chemistry is depleted, causes wear in the engine.
An "ideal" oil is easy to pump (i.e. a 0W or 5W) and has more than enough bearing protection (a 30 or 40.) So a 0W-40 will be really robust - knowing full well that a 10W-30 was factory fill at one time. I have always like the 5W-40 grade because there are some minor technical formulation nuances I prefer.
In your mind, please separate the first and second number. A 5W-20, 5W-30, and 5W-40 will all have the same SAE viscosity performance at low temperature, regardless of base oil type. If you want to argue that a synthetic 5W-xx is better than a mineral 5W-xx at low temperature, then you have lots of automotive, oil company, and chemical company engineers and chemists that will side against you. Same for 0W-40, 5W-40, and 10W-40 at high temperature. The reason the SAE viscosity grade system exists is because all the "stakeholders" have agreed that the engine can't tell the difference.
A multigrade oil is made by selecting the base oil to meet the cold temperature properties, i.e. the 0W or 5W tests. Then a "viscosity index improver" additive is used to thicken it at high temperatures to meet the 100C requirement. So multigrades are thin oils made thicker, not thick oils made to behave thinner.
In the early days of multigrades and in mineral oil formulations, a 10W40 grade might have been 12%-15% VII. These additives are long chain polymers that can break down, shear, or polymerize into goo balls. The result would be an oil too thin, or an oil that formed deposits early in life. VII is much better quality today than it was years ago.
Synthetic base oil has a naturally high viscosity index. That means it uses about 2/3 to half the amount of VII that a mineral oil would. In fact, you really couldn't make a 0W-40 or 0W-50 out of a mineral oil (Group I or II, probably could with a Group III.) Therefore you shouldn't attribute the negative issues with a wide-split viscosity grade on a synthetic formulation.
All this viscosity stuff is actually a little more delicate to formulate because there are volatility specifications, must account for the viscosity of the additives, and what second base oil types you might want to use the additive package in (and read the data achieved in the first round of tests over to qualify the second base oil also.)
In the end, the additives are more important. While there are guidelines to swap base oils in and out of the final products, much more testing is required to swap additives in and out.
Thanks for reading this far.