It sounds like you've defined rent == profit, which suggests you need to adjust your definition of rent.
Economic rent is hard to wrap your head around; I defer to the Wiki:
In classical economics, economic rent is any payment made (including imputed value) or benefit received for non-produced inputs such as location (land) and for assets formed by creating official privilege over natural opportunities (e.g., patents)
In neoclassical economics, economic rent also includes income gained by beneficiaries of other contrived exclusivity...
In layman's terms I think it could be described as income for having/being rather than income for making/doing.
The wiki definition you quote is essentially what I was going off of. There are still terms that need to be defined, like "natural opportunities" and "contrived exclusivity," and that gets you right back to property rights.
Economic rent is hard to wrap your head around; I defer to the Wiki:
In classical economics, economic rent is any payment made (including imputed value) or benefit received for non-produced inputs such as location (land) and for assets formed by creating official privilege over natural opportunities (e.g., patents)
In neoclassical economics, economic rent also includes income gained by beneficiaries of other contrived exclusivity...
In layman's terms I think it could be described as income for having/being rather than income for making/doing.