Without knowing exactly what data the district had to work with, it's difficult to speculate too specifically, but I can add that ADHD is one of the disability categories (under health, typically) that does not require a diagnosis for a 504. OCR has established that you just need some of the relevant symptoms (e.g., inattention, impulsivity, organizational challenges). This makes it a convenient avenue that some districts have latched onto for qualifying students when the documentation for their putative "real" disabilities isn't available.
As to the IEP, the flowchart for eligibility is this:
1. is there a federally-recognized disability? If no, the process ends with a finding of no eligibility. If yes, the process continues.
2. Is the student making effective progress? If yes, the IEP process ends with a finding of no eligiblity, but may be referred to the 504 process. If no, the process continues.
3. Is the lack of effective progress a result of the disability? If no, the IEP process ends, but may be referred to the 504 process. If yes, the process continues.
4. Does the student require specialized instruction (skill remediation, not just accommodations) to make effective progress? If no, the IEP process ends, but may be referred to the 504 process. If yes, the student is eligible for special education, and an IEP is developed.
As you can see, there are several points at which a finding of no eligibility could be made, some of which have different standards in different states. For example, the question (#2) of effective progress is defined as grade-level achievement in some states. (Not fabulous for 2e learners.) Unfortunately, if you're in one of those states, there's not a lot you can do about it. Another common effective progress obstacle is that some districts mistakenly will not consider social, emotional or executive function development in the discussion of effective progress, which is a misreading of the federal regs.
It's also worth reflecting on what the nature of the neeeded supports really would be. If there is not a teachable skill to remediate (#4), then the team cannot develop goals for the IEP. In some cases, the appropriate document is actually a more robustly-constructed 504 (which, btw, can include related services, such as speech/language therapy, counseling, ot, or pt). If there is a specific remediable skill needed, then clear articulation of that skill should guide the development of the IEP goals, (objectives, if you're in a state that uses these,) and service delivery.
Given that your district qualified him for a 504 based on concentration and is giving him accommodations which are essentially executive function scaffolds, one might ask what evaluative data they had in the IEP eligibility process to identify the presence or absence of deficiencies in executive function skills, and, if adequate measures were included, and EF needs were identified, why they did not determine that there was a need for specialized instruction in EF (which, btw, is what the private SLP is doing by working on organizing his thoughts for written expression). There should not be a question about #1, since they wrote a 504, which requires a disability. So it's somewhere in #2-4 that they made the no finding. Unless you're in one of the effective progress=grade-level achievement states, I would guess that the question hinges on specialized instruction, and what skill would be remediated. (One might propose that the skill is EF.)