While I don't usually recommend using the false grip on flat or decline bench pressing, it does provide a huge amount of benefit for pressing around shoulder pain on an incline. Over the last decade, we've transitioned more athletes to predominantly incline pressing with the false grip in order to nail the pecs and shoulders while minimizing unwanted joint stress.
Un-racking the bar with a stabilized false grip can be a bit tricky, and it's no surprise that many training-related injuries actually happen as a result of poor setups as opposed to the actual lifts. So use a spotter to help guide the bar out from the rack and back into the rack after the set.
Practice the setup on all angles including slight and high-angle presses and develop the skill of stability at the shoulders in a neutral position. If you've been pressing with a traditional grip, the false grip may force you to lower your weight at first, but as your skillset increases, your numbers will eventually surpass your previous bests.
Sparing the joints and targeting more musculature bodes well for long term strength, power, and aesthetics.