"Minimize thrust output" is not a great constraint - the most efficient landing would run at 100% thrust to minimize the time spent thrusting against gravity. Another thing to take into account, which is that rocket engines have a minimum throttle setting, which for the Raptor is 40%.
Yes but if the 100% thrust exceeds the mass of the rocket you need to exactly offset your momentum so that you reach zero velocity exactly as you touch down, which requires very precise timing and doesn’t leave much margin for error. Time it too late and you RUD. Too early and you stop short, run out of fuel, and then also RUD.
At least for Falcon 9, this is true even at minimum thrust. They don't have the ability to hover at all. The rocket is so light when empty that the thrust of a single engine at minimum throttle during landing is enough to make it take off again. That's why they do a "hoverslam" or "suicide burn" as you described. I assume they still throttle down as much as they can in the last few moments to minimize the impact of imperfections in the landing.
They likely just target a trajectory that falls at the center of their throttling capability. The controller can then handle a limited amount of trajectory error by throttling up or down. I.e. if they can throttle down to 70%, then they would compute a trajectory that would nominally require 85% thrust.
