If there is a problem, its impact is likely to become more apparent as he moves up the education system, particularly at secondary school, where the demands on organisation become greater; what with moving classrooms and different teachers; more complex, abstract language and concepts; and being expected to produce longer work by themselves!
IMO, an assessment would still give a true picture, because children change schools regularly and undergo assessments in their first year - the behaviours this assessment would be looking for, should exist across several environments?
Stigma is the least of anybody's problems. The behaviour is already there, and which is better for DS - for the school to put it down as "bad behaviour", naughtiness, forgetfulness, disorganisation, untidiness, disrespect, poor parenting, whatever; with all the impact on self esteem or for him to be able say "I have ADD, and that is why I struggle with some things, because my mind works differently"?
Personally, I would prefer the label of ADD, because then the locus of blame is not put on DS and yourselves; there is a disorder in executive functioning, which is nobody's fault. As its recognised as a disability, DS will be entitled to the help he needs and protected from disability discrimination by the Equalities Act. If he broke his leg, would you say he can't have crutches, because the school won't like it?
(DD2 has ADD, diagnosed at university - she was ok at primary, but really struggled all through secondary and while I knew there was something wrong from age 12, I did not know what. She did not struggle any less, just because we had no name for it and being called "scatterbrained" or told off by her teachers, and ignored by her friends, when she could not keep up in conversation, did nothing for her self esteem!)