Well I suppose that’s one interpretation but I think it misses the forest for the trees. Everything I’ve come across has discussed how it exacerbated trade imbalances due to a relatively lower proportion of Americans living abroad than citizens of other nationalities. Americans want to trade with Americans, like Chinese want to trade with Chinese. So having Americans abroad in bigger numbers helps for trade which I think would have a much bigger impact than some small reduction in labour costs locally.

Why would the multinationals that set the agenda in the US care about any of that?

Most of them make more money abroad than they do in the U.S. Setting up foreign branches is critical for their ability to reach new markets. Most of these branches are setup by U.S. citizens or permanent residents who have a deep history with their U.S-based employer and often personal connections with the executives at the home office. It is much easier to retain and pay these people if the bureaucratic hoops for retaining American citizenship abroad are less, and they will be less tempted to jump ship and set up competing businesses that jumpstart a foreign country's industry if it's convenient for them to stay American citizens.

Once they've set up the foreign branch, they don't really care if they retain them. Most foreign branches aren't staffed by Americans. This is also a 1 time thing in each country that they start a branch in, not an on going concern.

>they will be less tempted to jump ship and set up competing businesses that jumpstart a foreign country's industry

The number of instances of this happening has to be close to zero.

In my experience, something else happens - someone moves abroad to set up foreign office, then jumps ship to a competitor to set up a foreign office for them too.

None of this impacts a foreign employer’s employment of an American.