If I get a positive response on a Covid-19 test for the purpose of travelling to the USA, and then do another and get a negative, can I use that one?

There's a saying

A person with one watch always knows what time it is.

A person with two watches never knows.

The basis of this saying is that if you have two watches you don't know which one is correct.

In your case you posit that if the first test shows a positive and the second test a negative, then you should go with the negative test. The trouble with this logic is that you don't know which test gave a false result. Was the first test a false positive? or was the second test a false negative?

Ideally if you tested positive with one test, I'd want to see multiple, consistent negative tests before assuming that the first test was a false positive.

Pragmatically I would treat a positive test as meaning you are infected and that you should self isolate, no matter what a later test says. While it may screw up your travel plans, it does the most to protect your fellow travelers against potentially being infected by you.

As an example of your logic, (and not making a political judgement), prior to the (1st?) US presidential debate for the election, Trump apparently tested positive and then on a subsequent test, tested negative. The negative test was the assumed state, yet a short time later he was in hospital getting treated for Covid.

This may depend on the country you are coming from, but in general no. I've written the answer with the example of the UK, but similar rules apply elsewhere.

If you test positive on a pre-departure test, you must follow the required next steps for someone who tests positive in the country you are in. In the UK that would mean self-isolating and getting a PCR test for confirmation. If you test negative on the PCR test you can stop self isolating. I am not aware of any country where the appropriate next steps are to take further lateral flow tests until you get lucky.

In many countries, Covid-19 is a notifiable disease, meaning that the organisation which tests you is obliged by law to report positive tests to the local health authorities.

I'm planning to travel to the USA and I feel concerned about the false positive rate of antigen tests for Covid-19.

You should feel concerned about the false negative rate, not the false positive rate. It is shocking that any country allows intercontinental travel without a negative PCR test.

False positive — worst that can happen, your travel doesn't occur. Nobody gets hurt.

False negative — worst that can happen, you bring a new variant to another continent and that variant kills a million people. More likely, you infect some fellow passengers or people in the destination country with an already-present variant, and you hospitalise just a handful of people, some of whom might recover without long covid.

Can you travel after a positive test? Maybe. You can do many things that are either illegal or immoral or both.

Should you travel after a positive test? No, most certainly not. Even if your positive test is followed by two negative tests, JohnatanReez's answer illustrates there's still a 1.3% chance you are actually positive, which makes travelling entirely irresponsible. If 400 passengers reason that way, that would mean on average 5-6 infectious passengers boarding a plane. How many are infected after a 10 hour flight is anybody's guess. FFP2 masks are great, if worn continuously and perfectly by all; that condition may not be met on an intercontinental flight, and 10 hours is longer than what those masks are designed for — so it would be wrong to think your infection state is not important as long as you're wearing a mask.