There's a lot more it to than that. These systems are not only absurdly expensive (Patriots [1] costs about $1 billion for the system, and $4 million for a missile - probably more nowadays on both accounts), but also entail extensive training, operational, repair, and general logistic issues. A single Patriot requires a skilled crew of ~90 to operate with a peak theoretic range of < 100 miles.
And the systems are not invincible. When Russia either damaged or destroyed one recently in Ukraine, it seems they did the most obvious strategy and simply drained its battery (which seemed to hold ~32 missiles) before going after the system itself. Finally I'd also add in manufacturing and other issues. The manufacturing of alot of these systems has major practical raw material constraints (to say nothing of the manufacturing itself) when you talk about doing things at truly large scale.
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That said, I agree this idea sounds more like a MIC money grab than anything else. You'd need a ridiculous number of these systems to provide sufficient coverage, it'd entail completely scrapping the Outer Space Treaty creating an overt weaponization race, and also create some really juicy targets for relatively cheap anti-satellite missile systems. The reality is that there is no such thing as a perfect defense because it's just so much easier, and cheaper, to create a +1 version of offense, in the scenario of a peer to peer battle.
