...many aircraft have been lost following the ole chestnut "take it into the air sort out the problem and land"...
Any notable examples of the above, references and/or links? I say this because I know of many accidents caused by post-V1 rejects but very few where the outcome would have been improved by an attempted stop. The statistics appear to be very much against you in this scenario.
Taking the recent accident at MEL, I don't think things would have been radically improved if they'd aborted when they realised something was wrong (about 600m from the end of the runway). The report on the AMS overrun reveals that the P1 *briefed* about a >V1 reject, then performed one on an aircraft that was perfectly capable of flight, destroying it in the process.
...I like to think given the option most pilots would choose to have all options open.
Oh yes. We like options. However, experience has shown that exercising this particular option almost always results in a poor outcome. There's nothing to stop Captain X making this decision if he thinks that it's the safer option than continuing with the takeoff. All I can say is he'll need to hire a bloody good legal team if it goes wrong...