So which one is it? Is double taxation a real thing where you're taxed up to the sum of both countries' tax rates for anything above that exemption number, or do the treaties in place make it so that in practice you're never getting double taxed?
Many countries recognize foreign taxes paid on foreign sourced income, and the US is not an exception. You have the foreign tax credit (FTC) available in this situation, which will drastically reduce the US tax exposure. This is regardless of any treaty as the US tax treaties usually don't apply to US citizens.
The double taxation elimination clause usually does apply, but the FTC covers it.
Where can you be truly double-taxed? At the State level. Some States (like California) don't conform to the Federal law and don't recognize the Federal treaties. Most treaties don't explicitly include State taxes, so if you're considered a resident of that State you'll end up paying taxes to it on your full income regardless of any foreign taxes paid. That is why it is a good idea to change residency to a State with no income taxes, or at least conforming to the Federal law, before moving outside of the US.
Foreign earned income exemption is another tool to avoid double taxation, but it only works for earned income. It doesn't apply to passive income like capital gains or rents, so if you have foreign source for that income you'll only be able to use FTC.
In some cases the US taxes what other countries don't. Pensions are a good example: while the US has explicit treaties with some countries to cover that topic (e.g.: Canada), with most there are none, so employer contributions to (usually locally tax-free) pension/retirement savings are considered current wages by the US. This leads to very problematic double basis accounting - you have no tax basis under local law, but you have some under the US law, and you need to account for that for the time when you start getting distributions. Similarly with many other tax-free/tax-deferred compensation schemes in many countries.
Last but not least - the FTC calculations. This is not a 1 for 1 credit. You don't get credit of $1 for each $1 of the foreign tax paid. There's a certain calculation, for different income types. In other countries the same income may be categorized differently. So you may end up with lower capital gains tax in the US on something that's considered higher-rate taxed regular income in the other country and vice-versa, and because of that the FTC will not cover fully the taxes on one type, while you'll end up with unused credit on another.
Investments (like PFIC) are a another good example where you can easily be double taxed without any help from FTC. What's considered a trivial investment in a mutual fund in any other country (even in the US, if you invest in a US fund), becomes an extremely complicated and dangerous tax exposure if you're a US tax resident. Taxes on investment mutual funds outside of the US are at least at regular income rates for Americans, and can be much higher - while everywhere else it would be considered regular capital gains, or not even taxed at all.
The fact that the US taxes its citizens regardless of their residency is a pain in the a$$ for many who work or live out of the country. Even if you don't end up paying extra tax because of all the credits and exemptions, just dealing with this and paying tax advisers to prepare 100s of pages of tax returns is a tax on its own: it costs hundreds, if not thousands, of dollars just to be in compliance even if there's no tax owed.