When writing, what of these two mythologies should be used, in terms of quality and preservation of the original sources? (I am not listing other mythologies like Aztec, Mesopotamian, Caribbean, Polynesian etc because they are not well known or familiar to most readers)
It ultimately depends on your personal preference.
There is a wealth of lore from both Norse and Greek mythologies. I'd recommend you do some research on your own to determine which best fits your story. I think (in my uneducated opinion) that Greek myths are better known and thus have been preserved more deeply than Nordic myths. However, there are plenty of stories in each.
This might also help: https://viking.style/norse-mythology-vs-greek-mythology/
No one can tell you which better suits your writing, however. It all comes down to which you like the most and which you'd be more comfortable writing with.