I'm not going to denigrate the skills or physical fitness of soccer players. However......I find the idea that they are hyper-accurate when shooting at the goal very counter-intuitive. Isn't the goal something like 10 feet high and 23 feet long? I don't care how 'cat-like' your reflexes are, if the players were so fast and accurate with their kicks, from great distances, there's no reason not to have much higher scoring games. The net is HUGE!
And look at a soccer shoot out. All the goaltender can do is guess and jump a certain way (or stay still) and hope. They certainly don't have time to use 'cat-like' reflexes to stop the shot if it's moving at 80 mph and they have so much net to cover!
Now, I don't know what it's like in leagues with smaller nets.
The goal is 8 feet high and 24 feet wide. I played keeper in high school using those dimension and while it sounds big, it can get surprisingly small when the action is right in front of it.
As for your theory that a wider net means more goals should be scored, that's not quite right. Shots must be accurate for that to happen and with the more prolific forwards, this is often the case. For lesser players, this becomes a problem as many of their shots either go off target (missing the goal completely) or get blocked by opposing players or the opposing keeper. This isn't even discounting just how good or bad a defending team is vs how good or bad an attacking team is. If the defense is bad, but the attack is good then yes you're going to get a goalfest. If the defense great, but the attack is poor then yes you're going to be looking at 0-0, 1-0 or something like that. If both defenses and both attacks are good, then you have a thriller on your hands...I'm talking 2-3, 3-1, 4-4 scorelines etc.
As for the ability of the goalkeepers themselves. They rely primarily on their reflexes, just as hockey goalies do. The big difference is that a keeper must command a larger area, not just the goal mouth. They often can be seen higher up just outside of their box, barking out orders to their defenders.
In PK shootouts, it's essentially a guessing game so you're right in that regard. It is a bit more complex than that though. Some keepers are just able to "read" the PK-takers body language and base their guess off of that. This is also something that the PK-taker does as well. If the keeper gives them even the slightest idea of which direction they are going, the attacker can choose to go the opposite direction. So as much as it is a guessing game, it's also a bit of a battle of wits.