Yes, your analysis is helpful. But a reminder- correlation is not causation. The fact that rich kids apply early- and the presence of many rich kids in the accepted student pile- does not mean that being rich got them in.
Being rich (whether plain vanilla affluent, two professional parents or mega wealthy) isn’t what got you in. Whether to Dartmouth, Harvard, or anything in between. It’s that being full pay is associated with all the other things that matter in admissions- excelling at your EC (easier to do with a parent driving you back and forth to cello lessons, picking you up at lacrosse practice at night, making sure you don’t have to babysit for younger siblings so you can take that research internship that Daddy arranged for you) etc…
There are plenty of full pay kids who think that’s the golden ticket to Dartmouth (et al) and end up at their safety schools. There are entire college advising industries devoted to this cohort- helping parents understand the “better” options which will take a B+ full pay student over an A+ full need student.