Hello and thank you for your question.
There really isn't a downside to having your child tested. You could just explain to your child that the tests are just to see how he is doing in general, so you don't worry him. From what you describe such as low self worth, or being hard on himself for making a mistake, doesn't in any way indicate that he has ADHD or Asperger's.
Does he seem to have trouble focusing or with attention span? Maybe something happened at school that has made him feel bad about himself. Have you asked him if anything occurred to make him feel this way?
Thanks for the additional information. I do think that from what you are describing, that you should definitely have your son assessed. He does seem to have some traits that are of concern and the only way to know for sure what is going on, is to have him tested. You can ask the school how to go about this or contact his pediatrician and have him assessed by a child psychologist/neurologist who specializes in childhood developmental issues. The good news is that he is young and with early intervention can make huge strides in development and in lessening some of these quirky traits. You also need him to learn how to accept mistakes and not feel like he is a failure, as they are simply part of life and learning.
So, again there is absolutely no reason or downside in having him assessed. Time is of the essence as the brain can be easily molded while it's still developing, and you want to take full advantage of that opportunity. Please let me know if you have any further questions, and please do click ACCEPT button so I'm credited for my help today. Thank you,