Given that most of the schools discussed here maintain a roughly 70/30 full-pay/financial aid ratio, chances are definitely better in the FP pile. If you apply for financial aid, you will complete and submit a Parent Financial Statement (PFS) to the SSS which will return an Expected Parent Contribution (EFC) to each school your child applies to, and each school will know if you have the ability to be FP. If the EFC indicates you can be FP, each school will expect you to be FP each year your child attends unless some financial circumstance seriously and demonstrably changes.
There are lots of threads here discussing this issue. We are an example of a family who did not think we could possibly be FP and applied for (partial) FA only for the EFC to determine that we did not qualify. Two schools contacted us on M9 (one day before decision day, M10) to say they would like to admit our son but only if we could we could be FP as they could not award any aid to us based on our EFC. We had 24 hours to decide. It was a gut-wrenching time in our house as pulling that trigger meant a draconian four years for us, but we did it, and (12 years later) have no regrets.
Bottom line: You don’t really get to decide whether or not to be FP if the SSS decides you can write those checks. Best to just apply FP outright as not all schools will call/move your child’s app to the FP pile if don’t qualify for FA. Don’t take that risk.