Re: the passport discussion:
The most critical difference people are missing is that the US, Canada, the UK, and Ireland (and perhaps others) do not conduct routine passport control on exit in airports.
Russia does (as does the EU/EEA Schengen Area and most other countries). A Russian citizen cannot leave Russia without presenting their Russian passport. If they present a foreign passport, they will be immediately flagged for not having a visa or a stamp. There is a formal checkpoint before getting to the international departures area.
A U.S. citizen can leave the U.S. without presenting their U.S. passport (regardless of what the law technically says). I have done it countless times. When I leave the U.S. to fly to Europe, I enter my European passport in the Advanced Passenger Information form. I show my European passport to the airline and at the gate. There is no government checkpoint. No U.S. government official asks to see my passport. If one did (e.g., random spot checks), I would show the U.S. passport.
As noted above, other countries do passport exit checks routinely - e.g., it is impossible to leave the country without showing the same passport you used to enter. If I am leaving Italy (for instance) and flying to the U.S., I would enter in my U.S. passport information in the Advanced Passenger information, show my U.S. passport to the check-in desk, then show my Italian passport for the Italy / Schengen exit check (the purpose of this check is to determine how many days you've spent in the country / area, and if you've over-stayed), and finally show my U.S. passport when boarding the plane. If I erroneously showed my U.S. passport on the Italy / Schengen exit check, I'd be flagged as an over-stayer or illegal immigrant and likely detained and questioned.
Actually you have had to show it to leave the US. In the US verification of passport and other documentation has been delegated to the airlines. Which means they have to verify you have the right documents to leave the country.
And in the US they don’t really get hung up on which passport you have. Other countries are different
They're verifying that you have the right documentation to ENTER the country you're going to because the airlines are liable otherwise. In most cases, if you're denied entry, they need to transport you back to where you came from. And they absolutely do care which passport you have if you don't have a return ticket.
Going back to my earlier example, where I'm flying from Europe to the U.S. and back, I need to change the Advanced Passenger Information in between my departure flight and my return flight. When I'm flying to the U.S., I need to enter in the U.S. passport info in the Advanced Passenger Information. Before the return trip, I need to modify the Advanced Passenger Information to my Italian passport.
I have forgotten several times, leaving in the U.S. passport information. If I try to check-in online, I get a "see an agent" message. The airline identifies that I'm a U.S. passport holder travelling to Europe without a return ticket (liability to the airline!), and they ask to see my visa or permission to stay in Europe. I then show the European passport, and they do a manual update.
As for different experiences: I believe not all airlines check passports at the gate. Some only check passports at check-in, and then page passengers who are carry-on only to check their passports rather than having everyone show passports at the gate. And I've been part of AI-enabled boarding, where your passport is scanned at check-in, and then facial recognition allows you to board a plane wiht no check of documents or even boarding pass.