I now see that 20 is indeed the maximum, that an example is easy to find, that my intuition was wrong. Also I see that there are at least two mistakes in my previous post (a typo and an omission); nevertheless,I think the meaning is clear.
Is it possible that a system of inequalities could solve the problem algebraically or contribute to its solution?
I assume that you found each number in [2,20] in the count of obtuse angles in the millions of 6-point sets you investigated. Have you any record of their distribution?