Part of the problem here is the home position at the far right x=96 and your desire to cut the material starting on the far left. Change the home position X to 0, choose 'Shortest Path' for the "Order" and 'Keep Current...' for the "Start At" and all seems to be just how you want it.
For that matter if that rectangle on the <Default Sheet> is the actual size of your material (why would you not set the job size to be that instead of 97x49 inches?) keeping the home X at 96" and doing the changes above also seems to be what you are looking for.
The home position is set to the parking position of the spindle and the rectangle is the size of the stock I was cutting from, I am often using offcuts and odd shaped remnants so I got in the habit of just nesting inside that shape as opposed to changing the stock size every time as I found it to be more cumbersome than just creating a shape. Changing the home position did seem to help with tidying up the cut order but it still will not sequence the cut in order of the lowest to the highest on the sheet, but maybe my definition of 'Bottom to Top' is different from theirs. What I would interpret as 'Bottom to Top' and what I have found to work the best is illustrated in the 'Vector Selection' picture in my original post. But I'm gathering that is beyond what ever algorithm they use to determine 'Bottom to Top'.
Your start point node, from what I see, is on the left side of each circle yet in your previous statement you say "the start point is crucial in making sure the piece is still attached to the main body of the sheet throughout the cut". Where do you think that start point should be and if it is different than the node that currently is you should use node edit on one to make sure it's where you want it then re-nest the circles?
To avoid having to create custom start nodes on whatever varied and odd shapes are nested I usually just move the start point to the appropriate location using the 'Move Start Points' depending on whether it is a climb or conventional cut, with the circles being a climb cut the start point worked just fine being on the top right.
I mean it all works and I can work around it with 'Vector Selection Order', was just trying to find a way to avoid having to select every vector in order on the 40-50 sheets that are sometimes required, and it often makes the 'Apply Template to All Sheets' gadget unusable, which is unfortunate as it is an amazing time saver when it works.