I'm afraid there could be a hidden math behind this.

As in: cost of a war with China vs. economic cost of China acquiring TSMC tech.

As long as a war is cheaper, the US is protecting Taiwan.

I'm no expert, but it usually comes down to something like this.

US loves expensive faraway wars

They love wars against small third world countries that are easy to win. War with China is probably the only one that US can actually lose.

Not in this lifetime. Unless you are talking about occupying China / storming their territory, that would not go well. But defeating China at sea and in the air and in other countries? They'd have no chance.

China has 5 times as many people as the US. Their technology is behind, but they are modernizing and having 5 times as many resources and no public backlash to wasting soldiers lives goes a far way.

Some US generals disagree with you, especially when it comes to fight about Taiwan: https://americanmilitarynews.com/2021/03/us-will-lose-fast-i...

As you can tell from the last few decades, winning wars is a secondary concern.

Since when does the US government care about costs exactly?

I believe China would probably aquire a pile of rubble.