It's been a very long time since I had to think about this but I seem to remember from my cfi days that for an approach to have straight in mins it had to satisfy 3 requirements;
A "normal" or approx 3 degree final approach angle.
The final approach path had to cross the approach end of the designated runway..
Final approach course within 30 dgrees of the runway center line.
I think it was somewhere in the Terps.