It's up to your DM
The free interaction in combat explicitly calls out that the DM may make additional rulings based on potential difficulty in this area.
Assuming your DM rules that dropping an item doesn't use up your free interaction with an object, then yes that specific bullet point can be mitigated for by using/abusing the action economy rules. This has the proviso, as Szega mentions in their answer, that your GM doesn't rule that picking up a weapon in combat is a special action requiring specific care.
It should also be noted that Jeremy Crawford [has posted on twitter]:
The intent is that letting go of something requires no appreciable effort. But picking it up does.
Otherwise no you cannot do this.
Is War Caster rendered useless in this case?
No
The main benefits of War Caster are the advantage on Concentration Saving Throws and the ability to cast a spell for an opportunity attack instead of a Melee attack.
The Concentration Saving Throw advantage makes it much more likely that a caster will maintain concentration after receiving 21 points or less of damage from an attack (since you only need to achieve a 10 in that situation). The tables below give the chance of success with a given ability modifier, at a particular DC (assuming a nat 20 is not an auto success).
If you are in melee with someone who has the Mage Slayer Feat, which would normally impose disadvantage on concentration checks for you, the gains from the War Caster advantage cancelling out the Mage Slayer disadvantage are equally as large.
\begin{array}{c|rrrrrrrrr}
\text{DC10 check} \\ \text{(up to 21 damage)}& \rlap{\text{Ability Modifier}}\\
\text{Check Type} & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5\\ \hline
\text{disadvantaged} & 16\text{%} & 20\text{%} & 25\text{%} & 30\text{%} & 36\text{%} & 42\text{%} & 49\text{%} & 56\text{%} & 64\text{%} \\
\text{regular} & 40\text{%} & 45\text{%} & 50\text{%} & 55\text{%} & 60\text{%} & 65\text{%} & 70\text{%} & 75\text{%} & 80\text{%} \\
\text{advantaged} & 64\text{%} & 70\text{%} & 75\text{%} & 80\text{%} & 84\text{%} & 88\text{%} & 91\text{%} & 94\text{%} & 96\text{%} \\
\\
\text{DC15 check} \\ \text{(30–31 damage taken)}& \rlap{\text{Ability Modifier}}\\
\text{Check Type} & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5\\ \hline
\text{disadvantaged} & 2\text{%} & 4\text{%} & 6\text{%} & 9\text{%} & 12\text{%} & 16\text{%} & 20\text{%} & 25\text{%} & 30\text{%} \\
\text{regular} & 15\text{%} & 20\text{%} & 25\text{%} & 30\text{%} & 35\text{%} & 40\text{%} & 45\text{%} & 50\text{%} & 55\text{%} \\
\text{advantaged} & 28\text{%} & 36\text{%} & 44\text{%} & 51\text{%} & 58\text{%} & 64\text{%} & 70\text{%} & 75\text{%} & 80\text{%} \\
\\
\text{DC20 check} \\ \text{(40–41 damage taken)}& \rlap{\text{Ability Modifier}}\\
\text{Check Type} & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5\\ \hline
\text{disadvantaged} & 0\text{%} & 0\text{%} & 0\text{%} & 0.25\text{%} & 1\text{%} & 2\text{%} & 4\text{%} & 6\text{%} & 9\text{%} \\
\text{regular} & 0\text{%} & 0\text{%} & 0\text{%} & 5\text{%} & 10\text{%} & 15\text{%} & 20\text{%} & 25\text{%} & 30\text{%} \\
\text{advantaged} & 0\text{%} & 0\text{%} & 0\text{%} & 10\text{%} & 19\text{%} & 28\text{%} & 36\text{%} & 44\text{%} & 51\text{%} \\
\end{array}
I would also note that the Resilient feat you mentioned in your question does not give you advantage on the Saving Throw, simply proficiency in it. Proficiency varies drastically by character level, making the early level payoffs on concentration checks not as good, and the high level payoffs are of a similar order of magnitude as the advantage from War Caster, depending on your starting point.
In addition to this, if the player already has proficiency in Con STs from other sources they can still get good value out of the War Caster feat, while they won't from Resilient.
So the Resilient feat does not replicate the advantaged concentration Saving Throw feature of War Caster.
The Opportunity Attack (OA) modification enables casters to cast touch range spells as an OA, some of which can be particularly problematic for the recipient. Contagion, Bestow Curse and Inflict Wounds are good examples of this.
There are also combat situations where you may not want/be able to drop your weapon and pick it up again. For example, if you are grappled or restrained the potential for your DM to rule that picking up a weapon requires a full action is increased. In particular the illustrations in the PHB of the Grappled and Restrained conditions give the strong impression that certain free actions might be difficult when these conditions are applied, and thus require a full action. War Caster enables you to cast spells with a Somatic component in those situations without having to potentially use up a full action.