But if you were taking off from the Salt Flats with a 'runway' many, many miles long, you could ignore V1 and stop any time up to just after Rotate Speed?
Not necessarily. There is a lot of kinetic energy in an aircraft at Vr. The wheel brakes must be able to absorb this energy without overheating or catching fire. Thus there is a maximal possible v1. Of course, this assumes maximum braking so if you are braking carefully you could stop from higher speeds. But then you would be in unknown territory as regards stopping distance. (Wheel/brake fires are not unheard of in emergencies...)
And even then, in the event of, say, a serious engine fire you might consider chopping the power and landing straight ahead rather than continuing flight?
Sure, but again this is all very hypothetical. A transport category aircraft is moving really fast and runway will disappear behind it quickly. Since you have no figures, you would need very very large margins to do this safely.
It is not like the spamcans we PPL types usually fly where you could safely make several take-offs/immediate full stop landings in sequence on a 3000 m runway.