He didn't win keeper of the year award. Alisson did (PFA Team of the Year Awards).
De Gea won the Golden Glove, an award given for clean sheets. And cleen sheets is definitely a team effort.
For your first question, it's hard to accurately quantify which is more important.
It is very telling thoug, that Ederson, who isn't a great shot stopper, is widely considered one of the best goalkeepers in the world. Same goes for Onana, and to a lesser degree, keepers like Ramsdale.
Like you said, it's becoming very clear that the top teams and managers value ball control, sweeping and cross claiming higher than just shot stopping.
Perfect example: When Pep joined City, they had Joe Hart in goal, who was considered a fantastic shot stopper, and was a star player in that team. The first thing Pep did was replacing him. He then brought in Claudio Bravo, who also was a good shot stopper. Pep soon dropped him also, because none of them had the skills required to implement his style of positional and possessional control. As soon as Ederson came in, everything started to fall into place.
So to summarize: It's evident that shot stopping is considered less important than the other factors in the modern game. That's not to say it isn't important -- it absolutely is.
But most top goalkeepers are good enough shot stoppers, including De Gea. It's a basic skill for a goalkeeper.
Yes, De Gea is nowhere near as good a shot stopper as Costa and Allison. But shot stopping is not his big problem. We only conceded 6 more goals this season solely due to his shot stopping. While not good, that is not a catastrophic tally by any means. What definitely is detrimental, is his other areas that seriously holds us back, not only defensively.
De Gea won't sweep. This makes every ball over our defense potentially dangerous, and can lead to more chances and goals conceded than necessary.
De Gea won't maintain a high line, which means our defense has to stay low, instead of joining the midfield more often. This in turn makes it harder for us to feed the ball into our attackers. So by having a keeper glued to his line, we will not only concede more chances, we will also create less. This is one of the reasons why our strikers often seem to have the proverbial "graveyard shift".
De Gea won't claim crosses. This not only means that every cross is more likely to result in chances and goals against. It also means we greatly reduce our chances of gaining control of the ball, and start a new attack.
De Gea is very poor at passing through a high press from our opponents. Instead he will boot it long on average around 20 times each game, which is quite easy for opponents to recover. This also results in a lot more chances and goals against, as well as less chances and goals created for us. We can't possibly build an attack every time the ball is given away. It means we spend a needless amount of time defending, and chasing after our opponents in the midfield. No wonder we seem to pick up so many injuries, and no wonder we seem to run out of steam after 60 minutes. Our players are spending way too much time doing intense running and pressing, because we give away the ball 20 times each game.
And finally, De Gea's shot stopping has declined. Like mentioned earlier, it has resulted in him letting in 6 unnecessary goals this season. That is by far his least serious problem, and we could manage that, if his other areas were a lot better. The big problem with this, however, is that several of these goals has led to lost points as well. It's one thing to make a clanger when we are already losing 2/3-0. But when he makes those clangers when the scoreline is tight, it directly results in incredibly valuable points dropped. So in that sense, his shot stopping is not good enough either. But it's certainly his most "manageable" deficiency.
I know I couldn't precisely answer your question, as it's very hard to quantify exactly how important his shot stopping is. But every evidence available points to it being one of the least important skills, only in relation to the other skills, and also remembering that virtually every professional goalkeeper is a decent shot stopper. It's just levels of exactly "how good". The very best shot stoppers will prevent those 6 goals, and maybe add another 6 by saving shots that really are "impossible" to save. So at most, we're talking a 10-12 goal swing by only counting shot stopping. That difference isn't negligible, but everything points to that the other areas will net you a bigger positive than that.
And finally just to clarify, those numbers aren't my estimates. They are based on the xG stats data that measure the probability of saves and goals. The less scientific "eye test" also backs these factual stats. At least if you're just partially versed in modern tactics, and especially the latest revolutionizing of older philisophies like Total Voetbal and Gegenpressing principles that permeates every top team and manager's philosophies today. Real Madrid is an outlier here, with their counter attacking style. And they got found out big time by both Barca and City this year. The main reason why Real Madrid have been so syccessful, is because they are fast, and have excellent players all suited to that style. But Pep all but proved in that semifinal which system is more likely to dominate, and destroyed Real Madrid to the 'nth degree.
Brighton has adopted this style of football, with incredible success. The only thing holding them back is their limited budget, and lack of the very, very best players. They can compete with anyone, with a squad of players almost no one had even heard about before they joined them. That's insane, and just goes to show you how effective those tactics are.
And like mentioned and so effectively proven by data in this thread, the goalkeeper is the very crucial foundation to even begin to implement that style of play. You can perfect it with great outfield players. But you can't really begin to effectively implement it until the right goalkeeper is in place. That's exactly why we had to abandon that way of playing after being humiliated by Brighton and Brentford in our first two games. After that, we just started to let De Gea boot it, lowered our defensive line, and focused more on reverting to counter attacking similar to how we played under Mourinho and Ole, albeit with a much improved and aggressively implemented pressing system than we had under those two.
Any top manager who isn't stuck in the '90s will tell you the same. And they often do in their press conferences and interviews. Especially Pep, he loves to talk tactics. No wonder he does, he's near well tweaked and perfected Total Voetbal. So that's why Jason Steele now plays for Brighton, who recently played in League Two. He isn't considered nowhere near the best goalkeeper in the world. But he can pass like there's no tomorrow. That's why Arsenal bought Ramsdale. That's why keepers like Joe Hart had no chance of playing for Pep, despite of his great shot stopping. And that's why De Gea needs replacing, and has long since been completely dropped by not only one, but two different managers for Spain's national team.