This is a subtle goods type thread, impressive. I haven't seen this quality of the subtle goodness in a long, long time. However, taking all bs out of the OP, first just look to other teams in all scenarios:
1) What teams have cap space next year (reasonably, not by magically dumping all their players) to offer Kawhi a max contract.
o Within that subset of teams - how many have won les and/or are le contenders with Kawhi next year and longer term?
2) Assuming the pretty far-fetched idea that Kawhi takes the QO, what teams the year after have cap space (reasonably, not magically by dumping all of their players) to offer Kawhi a max contract.
o Within that subset of teams - how many have won les and/or are le contenders with Kawhi next year and longer term?
So even taking the Spurs out of the equation totally and just looking at things outside the Spurs, the answer to those questions leaves you with a pretty damn small subset of teams. You have to assume, based on the OP, that winning is a huge factor in these assumptions which narrows things down quite a bit. If it were just about money, the Spurs can pay him the most and all of the OP is moot.
Since the OP is assuming winning is possible, there are very few teams with both the cap space & roster in place that have proven they can win a le in the last 15 years and/or that can seriously contend while having max money too.
Then when you look at the Spurs there are many obvious reasons as to why he would stay logically, outside of money (which the Spurs have the edge).
o Kawhi's own words - he has flat out said he would likely be a lifetime Spur. That he values playing for one team. That carries the most weight since it's the most credible known source to date.
o Kawhi's comfort level - He, from his own words and observations of others, seems to be a low-key person. SA has proven to be about as good as it gets for players with that demeanor (see: Duncan, Tim).
o Coach: He has the best coach in the NBA.
o System: He has a system he is comfortable in and growing in. We've seen in the past, learning new systems can be difficult. Many players (histrocially) seem to value stability and systems they know work for their skillset.
o Talent: Even sans Tim/Manu, the team still has very good talent and some wiggle room to add more.
o Track Record: No team in the past 15 years has a better track record of winning and sustaining than SA.
o Loyalty - this is the most overrated term in sports, but there probably is a bit of loyalty that factors in, especially when the Spurs can pay the most money too.
So, in conclusion, when you look at both external (most important) and internal (still very important) factors, the underlying premise of the OP seems highly unlikely. Possible? Yes. Likely? A resounding no.