OK, the zombie master revives me. Beware.
First. The diagram is NOT doing what we are doing to the block (if it were another person). The model shown is a translation (against sliding friction) without rotation --whereas what we are (usually) doing is fundamentally a rotation (every several different types and cycles) but rotations nonetheless. How do you move a refrigerator single-handedly?
Vectors are harder in many respects because they involve an abstract force with an acceleration term, and are hard to "see" when acting in more than one plane. Moment just involves distance and mass, and rotations from one plane to another are relatively easy to "see." Rather than using the method of vectors -- use the method of moments to analyze it and see what you get. The most efficient method of moving any mass is by rotations -- either directly or indirectly.
Second-- the resistance in your scenario is from ground friction. Think about how to defeat the ground friction of the mass using cycles of motion. Think about how without pushing on anything you get a swing to swing higher. It is a critically resonant pulse. Do that to the mass. People in some respects are easier because they are reflexively responsive to resonant pulses. What is going on in the body to do those things reflects what is being done outside the body by doing them.
You reminded me of what I wanted to add to my reply post to David. Simply why the whole point of either of your approaches fails...
Aikido is not one inanimate object moving another inanimate object. Nor is it one person moving an inanimate object. Aikido and Aiki is a holistic approach to one's entire universe where there is no movement between things as there is only one thing in any given state of existence. The moment that state ceases to exist is the moment where Aiki ceases to exist. What takes place at the moment of Aiki (practically viewed in physical confrontation) is a fundamental shift in the consciousness between two animate objects whereby the distance time and functional delay between the two objects is constantly approaching zero at the speed of light.
...best in training to all.