Because your low cost weapons will be intercepted by their low cost weapons.
The enemy gets a say in your plans, and is much more likely to have low cost weapons available then high cost ones.
The interceptor for a SHAHED is a quadcopter which doesn't need to fly as far or carry as much payload. Anyone can build this.
The interceptor for an Iskander ballistic missile is a Patriot interceptor: literally nothing else can successfully stop it reliably. Only the US can build this.
If your attacking systems are cheap, then the enemy can field just as many: Russia has a lot of drones in Ukraine now too, they were just playing catch up.
"The next war" won't have surprise drones as a problem, it'll have highly developed and optimized drone and counter drone systems.
Not quite the scenario from my parent. They said "high cost weapons taking out air defenses". Whatever the US equivalent of an Iskander would be (I used a Tomahawk as an example), the S-400 (i.e. Patriot "equivalent") would be used to defend against it at first/in his scenario.
If you want to turn it around, sure. Let's see how you'd want to take out a Patriot: high cost weapons, like an Iskander might try it? Costs about as much as a Tomahawk? Would need multiple ones, because the Patriot would defend itself against even multiple ones? But the Patriots cost as much and you want multiple interceptors for each Iskander sent its way?
What if I could send, for less money/resources, a drone swarm that also takes out the Patriot or at least expends more money/resources in interceptors shot from it, than I had to spend on the drone swarm?
I totally agree, it's "just a race". If I build an offensive drone swarm for $x, which is less than your high cost interceptors, you better build an "anti drone whatever thingie" (which might be anti-drone drone swarms) that's even cheaper.
But, thanks, essentially you're agreeing with me: Don't use your high cost stuff to take out SAMs and then use cheap drones. Instead, use cheaper stuff to swarm it out of existence. Just gotta be faster at being cheaper. Doesn't matter if you're the attacker or defender.
Zerg vs. Protoss.
"what if I just had a better system with no downsides or logistical costs that was also cheaper".
In reality: Ukraine reliably downs Shaheds using a mix of low cost technologies.
They mostly can't defend against ballistic missiles without high cost interceptors.
The Shaheds could do a lot of damage to the Patriot site if they could hit it...but they never get anywhere near it. That's the point: your low cost system does not have the capability to threaten the high cost one.
And in all this you've forgotten that attacking the SAM site is only being done to enable other operational objectives. The Patriot battery is defending targets many times it's value, including the logistics and launch sites of all those low cost defensive systems - or the logistics and launch sites of your own low cost offensive systems.
To the article: the Tomahawk missile costs about $2 million per shot. Assuming this article is true, the missile in question gives you maybe a 20:1 cost advantage...but can it do the same mission? Does it have the same range, or targeting, or precision? If you cannot fire these from the same range as a tomahawk, or they don't realiably hit targets, then they can be substantially worse for a much higher logistics cost to deploy (perhaps total: the truck blowing up because you had to drive it to the front line is rather a problem).