That makes sense . . . but . . .
The devalued dollar will cut the cost of US labor, US goods that are used to create the goods. Other countries input goods will go up in price . . . but . . . the net effect will be a more competitive product because of reduced US labor and US input prices relative to the foreign currency.
Make sense?
First lets look at the facts. Exchange rates and trade balances don't affect each other. Any logical system that proves reality wrong is useless. There are plenty of reasons why reality is what it is, but anyone can hear the reasons and proceed to come up with additional reasons why reality is wrong. That's where we really get lost.
Which is why the US trade deficit always falls when the dollar weakens....except for all those times the US trade deficit rises when the dollar weakens.
Make sense?
Not really. Dollar (any currency's) international exchange rate has no effect on the local cost of labor, because it's dictated by local tax and regulatory environment, e.g., minimum wage, take-home pay etc.; but it actually might increase other domestic input costs of goods produced for export, like domestically produced raw materials... in addition to more expensive foreign input goods and materials.
That's why the bulk of the problem is the cost of taxation and regulations (including "environmental"), not so much an exchange rate which only marginally affects overall balance of trade (positively or negatively, depending on the mix of foreign input / output in particular industries). In addition, input from raw materials produced from domestic sources but used in export output also get more expensive (due to internationally weaker currency) so the output gets more expensive and less competitive, whether internationally or domestically. So overall domestic production costs - greatly affected by politicians and bureaucrats, laws, mandates and regulations - have by far, by far greater input costs than the potential net benefits of output marginally affected by the weaker exchange rate.
Basically, weaker currency acts as a tax on domestic consumption and production so its potential benefits are ephemeral, at best. That's why devaluing / debasing the currency never works over long term, even if it serves sometimes as a cover (and a fodder) for politicians trying to justify their actions which increased the cost of production and consumption. In pushing dollar down, Bush got (not the only) bad advice from his weak economic team. Ronald Reagan's and Bob Rubin's policies of "strong dollar" served the U.S. much better in this respect.
Add to all of the above the intangibles, that while the free trade Americans / the West are used to and don't generally have a problem with buying foreign-made goods, in many developing countries people in much larger numbers may still choose locally made products, even if more expensive compared to the similar foreign-made goods, due to culture, habit, pride or other real or perceived advantages, i.e. cheaper exported goods will generally find easier market in the developed / "old" economies than comparably cheaper exports in emerging, less culturally diverse markets, which may negate any exchange rate "advantages" in those markets.
You can see the above empirically in the current account / trade balance of the U.S. generally becoming negative as the cost of domestic labor and production went up sharply, whatever the exchange rate was. To sum up, weaker currency is a tax (a de facto tariff on all foreign goods) imposed by government, which in the long term always punishes consumer, without any concrete (and usually temporary, at best) benefits to the exporters, with the government, in effect, choosing the winners and losers between industries.