There are several kinds of "big hole tournaments". The 8 inch, 10 inch and 12 inch cups CAN be missed, but not usually from 4 feet. 10 feet can be sketchy.
Then there are big hole tournaments in which the hole is basically the size of a garbage can. (Sometimes called foot golf) These are difficult to miss from anywhere on or even just off the green. You just hit it almost as hard as you can.
If you've ever noticed a putt from the fringe on any sizable length... the ball is in the air for the first 2 feet or so. Your putter has about 4 degrees of loft.
I will chip if I have more than 2-3 feet of fringe to roll through or if the the grass is leaning towards me... which brings up a whole new problem.
Well sorry but not a big fan of this one. Too many ‘it depends.’ How much green do I have to work with? Is it downhill or uphill? Downhill I’m using putter. A lot of green and uphill I’m chipping. But generally I use putter anytime I can.
Not enough information to decide. 3’ can be very different than 6’. The actual lie and grain orientation of the grass is important. As is the distance to the hole and the contour of the green between me and the hole.