So, I figured out why fishy numbers were arising in the calculation.
...
Expectation:
We should expect that when an orbit is put at an inclination of 23 degrees, and the orbit's parameters (a,e,inclination) are set such that the orbit is sun-synchronous, the time spent in the Earth's shadow (for access and observation of the satellite payload) should be the same throughout the year.
In other words, if an orbit is in the plane of the ecliptic, at 23.5 inclination, the satellite should spend some fraction f of its orbit in the Earth’s shadow, independent of time of year.
Problem:
We are not seeing this expectation fulfilled.
Solution:
There is a misunderstanding regarding how the orbit precesses with regard to nodal and apsidal precession.
...
There is one issue, though. When we do these calculations subtracting the for known sun-synched circular orbits though, my hypothesis isn't consistent.
So, given that the best solution would be to make precession such that dw +dtheta = 360, I wrote a script to extract those orbits, limit them to those with perigees above 400 km, and extract the one with the slowest angular rate (typically the farthest, most eccentric one). This is the candidate:
Further, I decided to find an orbit that minimized apsidal precession, made dtheta = 360, and existed at 23 degree inclination. This is the candidate:
The question still stands that even if a 23 degree solution doesn't exist due to this problem, there are still several orbits that were analyzed that, when combined or launched at the same time, can provide consistent coverage above 5 minutes for the entire year.