LGB,
I hear loud and clear where you're coming from, but, at the end of the day, it depends upon whether you're flying an aircraft which has GUARANTEED one engine inoperative performance from V1 (e.g. FAR 25 aircraft and their equivalent), or flying an aircraft that DOES NOT have guaranteed one engine inoperative performance.
For aircraft that DO have guaranteed one engine inoperative performance from V1, accident statistics heavily favour the continued Takeoff. For the aircraft which has no guaranteed OEI performance after V1, then it is up to the pilot's judgement - Is the available OEI performance and obstacle clearance adequate to ensure continued flight? If so, there should be no fear in continuing. If not, then the best "controlled crash" possible is the only option, and, if there is sufficient runway remaining, putting the aircraft back on the runway expiditiously sounds perfectly reasonable to me.
You pose the question - "why isn't there a procedure for this ...?" To begin with, there's an enormous number of variables, the most important of which seems to me to be the height above the runway that the Takeoff is aborted. We'd require a new term, "H1", to replace V1, being the maximum height at which airborne flight could revert to a landing. The computations would be enormous. I've done performance analysis for Touch and Go landings for heavy (FAR 25) aircraft, which is essentially a OEI landing converted to a OEI Takeoff, and that's a complex process. What you propose would be mind-boggingly more complex to prove.
I've been fortunate (unfortunate?) enough to have had 2 complete engine failures after V1, and one RTO from very close to V1 in my career. The RTO was a heart-stopper, whereas the continued Takeoffs were exciting, but relatively uneventful. Added to that are dozens of deliberate RTOs from near V1, and deliberate engine cuts (fuel off) at Vef during certification flying, and personally rate the RTO as 10 to 20 times as dangerous as the continued Takeoff. Statistics back up my gut feeling.
In closing, be assured that if you're flying an aircraft with guaranteed OEI performance after V1, and choose to land on the remaining runway, you won't have a legal leg to stand on, even if you don't even scratch the paint. For other classes of aircraft, it's up to your judgement, and the accident enquiry will demonstrate much "wisdom in hindsight" in examining your case.
Regards,
Old Smokey