Counsellors of state are always the monarch's spouse and the first four people in the line of succession who meet the following specifications: they must be British subjects of full age (21 years, or 18 years for the heir apparent or presumptive) who are domiciled in the United Kingdom and not disqualified from becoming monarch.

Those people are William, Harry, Andrew, and Beatrice.

"Domicile" doesn't mean where a person currently lives, so whether Harry means the qualifications is a legal issue that nobody really wants to explore.

It is enough to know that neither the King, Parliament and the majority of the people in the UK do not want neither Harry nor Andrew filling in for the monarch.

Charles did not want to explicitly remove anybody by name, fearing that Harry would act up (possibly sue?), so he suggested limiting the position to working royals, but that would have meant Catherine and Sophie would be eligible. I think Parliament didn't want to include any more people who weren't in the line of succession.

The quick fix was to add Anne and Edward for their lifetimes specifically to the list, with Charles giving assurance that he would only select working royals.

As to why limit to only working royals, the duties which require Counsellors of State to be appointed are those that are duties of the Head of State. It could be construed as disrespectful for a non-working royal to show up.

If a person isn't "good enough", is surplus to requirements, or just doesn't want to work for the full-time for the monarchy, then they don't get to act as Head of State at an event.