Do admissions care how close a TOEFL score is to the minimum?
The answer is "sometimes."

Some schools will not care what the score is, as long as you have the minimum requirement that they advertise publicly.

Some schools won't care, even if you don't have the minimum requirement (they may "waive" this requirement routinely).

Some schools will care: they may have one public requirement, and a higher (unpublicized) threshold that they really use in most cases. (Or: the school has one minimum and the department has a higher, unpublicized one.)

Some schools will care: the minimum is required to get your application looked at, but even if it is met they still consider the score in admissions decisions.