About the only time of would advise a false grip is when doing BB/DB (or even lateral raises in some cases) regular or reverse, curls. Why you would weaken an exercises muscle building potential, when using a heavier weight for DL's, rows, chins or other back exercises, does not seem logical. With a grip (false) that can allow a bar to roll off the fingers or just not be anchor well by the thumbs, before the full strength of that exercise is taxed, seems strange indeed. You may be just cheating yourself of the full benefit on most exercises. With the natural thumb grip you will have a superior vice grip style on the bar.
If the main plan is to improve the gripping strength, than a thumbless grip can be included in a program, along with regular gripping. Like the fore mentioned Farmers Walk (as suggested by TWT), DL's, chins, etc. With most of those lifts you are just holding (or trying to) the weight in place anyway. Lot of other ways to improve the grip/finger strength also. All this is intended for strongman stuff, not regular BB'ing. Good Luck.