One of the things I would avoid doing is begin the turn too early.
Looking at the approach plate, you know what your minimum safe altitude is in any quadrant.
If you are on the VOR or LOC approach and the ground equipment craps out, then I would immediately abort and begin my climb to safe altitude before I would begin my turn.
Of course, each approach and location is different. So, as you have pointed out, positional awareness that is granted to us by the moving map displays many of us have in our cockpits, enables us to make a more informed decision.
Along this line of reasoning, I believe that each pilot should have a minimum descent altitude (AGL) that he/she will descend to for any approach. My personal minimums are 500 feet AGL, no matter what the legal minimums are. IF I were flying into some challenging mountain areas, I might want my minimums a whole lot higher.
My attitude is (if there exists no emergency) I DO NOT have to land. I try to maintain a minimum fuel reserve of two hours. That way, I can travel at least 300 NM before I run out of gas. With the current METARS available during flight, I can often make my no go decision without even trying to "look see". At the price of gas now, it often is easier to make an approach to a VFR airport, sit out the weather (or spend the night) and continue on in the morning with full fuel.