They can't be examined at all. All we can do is use instruments to make measurements. And all instruments have limitations.
They are simply close enough such that our instruments can't tell the difference.
You're doing armchair science: drawing scientific conclusions from philosophical considerations. And you're doing it the way contemporary philosophers nearly always do it: by applying intuitions derived from the most up-to-date state-of-the-art knowledge of how the world operates that 19th-century physics has to offer. What you're saying would be perfectly reasonable if this were a matter of classical mechanics. But this is quantum mechanics, where human intuition is a trap. In quantum mechanics, we can tell
identical particles from distinguishable particles by a systematic difference in their behavior that has no classical analogue.
In classical mechanics, if you want to calculate the probability of an observation, you take into account all the ways it could happen, calculate the probability of each one, and add them up. If you do that in quantum mechanics you get wrong answers: you get predictions of probability that don't match the observed frequency when you do an experiment over and over. You have to use a more complicated procedure. You still take into account all the ways the observation could happen, but then you have to carefully sort the ways according to whether you could, even in principle, tell whether or not that's the way it happened. If it's possible in principle to tell whether it happened this way or that way, then you add the individual probabilities, just as common sense dictates, which is to say, just as in classical mechanics. But whenever there's no way to tell whether it happened this way or that way, then you have to instead calculate the "amplitude" of each way it could happen. (An "amplitude" is sort of like a square root of a probability, but it also has an angular component called a "phase".) You add up the amplitudes for all the ways the observation could happen (using vector addition, which means an amplitude at 1 degree phase can actually cancel out an equal amplitude at 181 degrees phase), and then after you add up all the amplitudes, you have to square the total since they were all square roots of probabilities. Needless to say, this complicated procedure does not give you the same answer as if you simply added up the individual probabilities.
That's the tedious part. Here's where it gets interesting. When two particles get close enough to each other, the Heisenberg Uncertainty Principle kicks in, you can't follow their individual trajectories with infinite precision, and when they come apart again they might or might not have traded places. Now, if there is any physical difference between the two particles, you don't need to have tracked their trajectories -- you could just examine the particles after the fact to tell which is which, and determine whether they got swapped. But if they're identical then there's no way even in principle to tell whether they traded places. So that means that when you calculate the total probability of a particle going into a detector, the basic rule of all quantum mechanics says you have to take into account all possible ways for it to get there, including the possibility that the two particles traded places -- and you have to do it by one of two different procedures depending on whether there is any way in principle to tell the particles apart. So we apply both procedures, and we get two different answers for the probability of the particle going into the detector. One of the predictions matches the detection frequency we see when we run the experiment over and over, and the other prediction is wrong.
If you do this with two helium-4 atoms, the assumption that they're identical gives you the right probability, and the assumption that there's some subtle difference we haven't yet learned how to measure gives you the wrong probability. That is why modern physicists believe the atoms are identical. It has nothing to do with our instruments' technical limitations.