There are many different scenarios included in the OP, by implication and it's not always clear cut which is the easier method. Getting the best out of your basic strength is a matter of smart use of your body. As
@Drakkith says, it's always good to 'lock' parts of your skeleton when possible and then use the strongest muscle set.
If you are pushing a heavy object, you will often be pushing downwards, against the ground which will increase the Normal Force on the ground. This will increase the friction force along the ground, which often means More Work Done when pushing and will lift a dragged object out of ruts and bumps.. That can explain many examples and it's a typical example in basic Mechanics test questions.
Then there is the way our bodies /skeletons work. We can lean backwards tp pull and use our skeleton as a lever (fulcrum being the feet). Some of the force on the pulled object comes 'for free' when using this leverage and also, much of the effort can come from the legs. (Muscles at the front of the thighs, in particular)
When pushing, there is no equivalent lever to use. OTOH, we can sometimes use our weight force for free, such as when pushing a car out of a ditch, using our butt and in that case the legs are doing the work mostly.
My strength is nothing like it used to be and I find it more and more useful to
think before trying to move something. No dead weight lifting for me, in any direction, I use the lever method whenever possible.
When you next see a deliver man with a heavy item on the stairs, why not engage him in conversation and ask him about his method. This would be preferably carried out when he is actually in the process of lifting. Other ways to make friends . . . . . . . .
