Honestly, I have always thought this to be correct; that Harry, if he is on an A1 visa, is in breach of it, because he is working for himself, and not for the UK government. I'm not so sure now though.
As I understand it, there are 2 things to consider, visa classification and visa status. Classification is A1, A2, tourist etc. Status is diplomatic or otherwise. The diplomatic status gives you certain privileges and immunities, eg immunity from prosecution.
I had a look at the section 41.26 of the Code of Federal Regulations on diplomatic visa status. Assuming Harry holds a diplomatic passport (I'd have thought that was a given because of who he is) then he is eligible for a diplomatic visa regardless of his visa classification:
Classes of aliens eligible to receive diplomatic visas. A nonimmigrant alien who presents a diplomatic passport or its equivalent shall, if otherwise qualified, be eligible to receive a diplomatic visa if:
(1) The nonimmigrant alien is within one of the following categories, irrespective of the classification of the visa under § 41.12:
(i) Heads of states and their alternates;
(ii) Members of a reigning royal family;
(Goes on to list others so eligible.)
All heads of state automatically receive A1 classification and diplomatic status regardless of purpose of travel - they do not not have to be traveling on behalf of their government. All other visa applicants who are applying for an A-visa must be able to demonstrate a valid purpose of travel related to government work. However, if I am reading the above correctly, as a member of the royal family, Harry doesn't need an A- visa to have the diplomatic classification. That said, I have seen commentary saying he will get the equivalent of head of state treatment (and A1 classification) because he is in the RF, although they don't explain how they make this leap of logic.
My own conclusion is that whatever Harry's visa classification is and whether he breached this, or whether he lied on his form about drugs, the diplomatic status he would have been afforded for (presumably) holding a diplomatic passport and being a member of the RF affording him a diplomatic visa, left him exempt from checks on his application by DHS in the first place, and immune from prosecution or other action against him now.
I could be very wrong, though!!