I've worked in fintech (lending specifically) and your claim that "there is a (very) large number of people who do this" is simply not true.
To get access to the half decent rewards credit cards in the US will at a minimum require you to have an ITIN. To obtain an ITIN, you need to submit an application to the IRS and provide a bunch of documentation related to your identity and foreign status. This documentation needs to be original or certified. If you don't use your ITIN at least once to file a tax return in 3 years, it will expire.
If you jump through the hoops to get an ITIN, as I mentioned, financial institutions usually have a database of private mailbox addresses. Technically these are called CMRAs - commercial mail receiving agencies. If you use a CMRA address when applying for a credit card, there's a good chance it will be detected. So you're going to need a friend or family member in the US to let you use their address. Also, if you apply with a foreign IP address, this too will likely be detected. Use of a VPN can trigger extra scrutiny.
The American credit cards that offer attractive rewards have the highest requirements in terms of credit score. If you have no credit history, you will not be approved for these. The average non-resident foreign national isn't spending enough every year on their credit cards to gain any meaningful benefit from a crappy American rewards credit card, or to spend years building up a credit history to get a better card.
As for people moving around the world, it's imminently easier for American citizens to keep their credit cards and get new ones when they become expats, especially if they don't change their addresses or set up alternative US addresses (with friends or family) before they move abroad.