How to “pedal a chain back on”

One of my roadies has the idiosyncrasy that when shifting to the big chain ring, the chain goes past it and falls off. One can pedal the chain back on, by pedalling slowly and moving the lever to the small ring. The chain almost always finishes up back on the small ring, ready to repeat the process again and again till the bike stops entirely. This unpleasant habit has slowly been worsening over the last few years, and I have straightened the big ring and set the limit screw. This last effort has resulted in less chain losses, but unpleasant chain rubbing noises in top gear in the big ring. When I get it adjusted precisely, the chain both rubs (because it isn’t moving far enough) and falls off (because it is moving too far). A simple solution is simply to ride another bike, but that is not quite cricket, not quite the done thing. (Following that logic to its conclusion, I would have to dispose of my entire graciously ageing fleet and purchase one (1) reliable and well-sorted bike which, after a few years, would start developing odd habits similar to those of my current bikes. Why make this effort?) One day I will sort it out. In any case, pedalling the chain back on will keep your hands free of chain filth, and save precious seconds better spent leaning on buildings after arriving at your destination.