This could work for falling, but check with your DM
Jumping is not falling
First, jumping is a form of active movement in the game, as explained under the movement rules (p. 190 PHB)
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
Falling is passive, it does not deduct its distance from your speed, so it is separate from jumping. So jumping onto a creature is not falling onto the creature, and the rules for falling, including the optional rules for falling in Tasha's do not apply to a jumping creature.
You could potentially do this with a flying creature like an Aaracokra, by flying up, then stopping to fly and falling. However, as this answer explains in detail, technically you cannot just choose to fall as a flying creature. So a DM may well counter your tactic of a technical reading of this rule with an equally technical reading of the flying rules.
Vertical space is not well represented in the rules. There is no height given for the space a creature occupies. But from common logic, you can only impact a creature by falling into their space, when you fall onto them in some way, and that means that you need to be at least above the creature before you start falling.
Moreover, you cannot enter the space of a hostile creature with your movement, so you would need to fly high enough to be over them when you start falling. Depending on how tall the creature is, this may mean you need to be at a height of at least 5 feet when you start falling. As falling itself deals no damage, the benefit of repeated falling onto the same creature would be very limited. As you need to fly up at least 5 feet, and move at least 5 feet to a new space, if you have a fly speed of 50, you could at best drop 5 creatures with this tactic in one turn if they all stand next to each other and are not too tall (possibly attracting opportunity attacks made at disadvantage as you leave their reach).
Does the creature you drop on fall prone if it takes no damage?
The optional rule says:
any damage resulting from the fall is divided evenly between them. The impacted creature is also knocked prone, unless it is two or more sizes larger than the falling creature.
I think looking at the paragraph in isolation, it is most accurate to read the paragraph as you suggest: the creature you drop on has to make a saving throw, and if it fails only takes damage if there is damage from the fall to be divided. Then, new sentence, wether it took damage or not, it also falls prone, the also referring to the failed saving throw and its consequences, which do not need to involve damage.
It however also is possible to read this in a larger context: damage from the fall is shared between the creatures, and then the impacted creature is also knocked prone. That is, the also refers to not just failing on the saving throw, but it refers to failing the the saving throw and taking damage. The context here is that you yourself only fall prone from falling if you take damage, that is, there is the general idea in the game that a fall has to be substantial enough to cause damage to make someone drop prone.
I think that is a less technical reading, but one that is more in line with how the game overall handles falling, and I can easily see a DM ruling this is how they run it, so before banking on using this tactic, clear with them if they are OK with it. Also remember this is an optional rule. A DM who feels you are explointing the wording may be incentivized to shut down the rule to begin with.