Fortunately, most aircraft are equipped with an alerting system that ignores minor failures when in excess of 80kts or so (the ones that get through are typically engine fire, failure or AFCAS - or whatever your aircraft calls it). Therefore, most of us stop for any alert below V1. Above V1, you stop at your own peril as we are now in uncharted territory - therefore this is only done "in extremis" such as control failure.
Liability doesn't enter into this - just do as you have been programmed (some might even call it training!). Trying to work out in split seconds whether you have enough runway to stop at a speed in excess of V1 will only end in tears.
Going back to the original question, if this happened at a low speed, yes I would stop, because I would get a warning and at a high speed I wouldn't, because there would be no warning. I don't let runway lengths change decisions, only the speeds if allowed by the performance tables.
Clear as mud?