Flu like symptoms are very common for so many types of infections. Flu-like symptoms are the body's response to infection. So you can't tell that he has N Fowleri just from his symptoms. Of course, this will not satisfy your HA.
You need to also recognize your tendency to catastrophize. Catastrophizing is when you jump to the worst conclusion possible based off mild symptoms or signs. It is part of HA and if you indulge it, you just keep yourself within the anxiety loop.
You need to stop researching symptoms, you need to stop googling in response to anxiety, and you should probably limit the amount of health/illness things you read in general. If you spend lots of time browsing these forums, you just allow others' fears to scare you as well, if you are deeply entrenched in HA.
There is so little of a chance your son got a deadly infection from a swimming pool that I would really advise against going to the ER as it is reassurance seeking behaviors that make the HA worse and worse.
Also, very young children just get sick a lot. It's how their immune systems develop. It could just be a coincidence your son has gotten sick, and you could be attributing it to the pool. The infection could have been developing well before you got to the pool.