Question

How might atomic radius have any influence on electronegativity trends or no influence on ionizatation

Answer 1

The radius is proportional to the ionization energy.

This is because electrons are drawn closer to protons, which have opposite charges and hence cling to them, in a small-radius atom.
If the radius is bigger, the electrons on the outside edge of the atom are not as tightly bound and are therefore more easily lost, requiring less energy to ionize.
Factors are more shielding (from core electrons) in the lowest elements of a family, allowing electrons to escape more easily. For those who are currently in a period, the effective nuclear charge grows as the period progresses (more protons, but no more energy levels, so the electrons are the same distance from the nucleus). This causes the electrons to be held closer together (smaller radius), requiring more energy to ionize them.