Money is normally a pretty big motivator but I am not sure it always fits every situation. Let's look at each case individually . . .
Sting and the other two guys weren't really getting along great and Sting wanted to go in a different direction musically. Sting was likely the majority contributor to the band, so it's not like the other two were going to get someone else to write the songs and sing and still call themselves The Police. The Police sold more than Sting did, so I am not sure the motivation for Sting was to get a bigger share of a much smaller pie. Similarly, there have been any number of times The Police could have reformed over the years and they only did so for one tour. There was always talk of getting together to record a new album and there was too much going on personality wise for that to ever happen. Sting had to be the one in charge so another album never happened. Even now, if they wanted a money grab, they could reform tomorrow and print money if they wanted to.
You make valid points so, out of my own curiosity, I took some time to research Sting's departure to see if he was better off with The Police or on his own and I'll share what I discovered below here :
The Police sold roughly 15 million records/tapes over 5 studio album releases in the US. Of course, if you count all the other countries it gets much higher but just for sake of a quick comparison, if we just look at the albums sold by The Police, US sales were around 15 million. Additionally, I'm absolutely positive those first few album deals were probably largely in the record company's favor so I would be willing to bet the record company (A&M) made the lion's share of profit off those first few albums. Regardless, profits still had to be split up further between the three of them beyond what the label took. You have producers, agents, engineers, etc.with their hands in that pie as well. My point is, even though they sold a lot of albums and made a lot of money, I'm not so sure they made as much off those first few albums when they were unknown. Even later on, when they became established, still, if the profits had to be split up between the band members. So, for example, The Police's best selling album was Synchronicity. So that album sold 8 million copies in the US right? Let's simplify it down and say that The Police band received $1.00 for every album sold okay? So $8 million / 3 = $2.67 million each. Even though Sting only sold 3 million US copies of his first album, still, if he received the same $1.00 per album sold, that's $3 million on his very first album alone and that's just for album tape sales. I would strongly argue that it was EXTREMELY lucrative for Sting to go out on his own when he did, it's not like he took a pay cut, if anything I'm sure he got paid more and I'm willing to bet he gambled on himself that he would be better off on his own when he split off from the band.
Sting, on his own, did 8 solo albums and he did about 15.5 million in US sales alone on those, and like The Police album sales, I'm not counting UK, Germany, Canada, France, etc. sales but that doesn't include the money from concert tours, merchandise, DVD's, etc. Once again, ALL of that went to Sting and I would wager his deal with A&M records was much more profitable in his favor since he was already an established brand at that point. Additionally, he also did 6 live albums and 9 compilation albums, again, all of that went to him, he didn't have two other partners to split the money with. The biggest point here is the royalties go to Sting and Sting alone. So if he sold however many million albums in the US in 1987, that's one thing, but then he can continue to get the royalties from all the times his albums sell from 1987 and on. I mean, hell, he may have sold another 10k albums per year from new listeners or sales on iTunes or song royalties for songs used in commercials, movies, etc. It's not just the number of albums sold from the year the album was released. There may be some resurgence in Sting songs next year where he sells a million copies of his first album because it was used by some famous Youtuber or something, again, all that royalty money goes to him, he doesn't have to split it between the other band members.
And as I pointed out though in a lesser point, in addition to the money, there's also creative, output and content control. If you wanted to do an album full of remakes, you don't have to ask your bandmates for their permission or input, you'd just do it. If you don't feel like putting out an album every year and you're in the driver's seat, then you'd put out an album whenever you felt like it. I think the other thing about is, is the freedom of splitting out from the band as well. I was looking at Sting's discography as I was writing this and he did a TON of one off projects with other artists and groups over the years plus his side projects in acting jobs here and there. Again, if he's in The Police band, I'm not sure he has that kind of freedom at the very least and at most, those solo projects, acting jobs, etc. would have surely caused friction between the band members because that would have taken time away from him making The Police albums, touring with the band, etc.