I don't want to get involved in an Orac vs John Farley debate so I won't!
However, when I use to teach about this stuff there were two categories of "Military Aid" that could be invoked.
The first was "Military Aid to the Civil Power", and had to be invoked by a Senior Judge at the request of the Police. The ever-present example of this is in Northern Ireland where the troops only work within the limits laid down by the judiciary. This emphasises the principle of the "Rule of Law".
The second was "Military Aid to the Civil Community". This category is not subject to Civil Law, but has clearly got to be conducted according to standard military procedures. There are lots of examples of this from flood relief to foot-and-mouth measures. There can be no question of a Service(wo)man not doing something because they don't happen to believe in it. And that must apply to the CDS as much as to a Lance Corporal in a fire engine. The Military personnel work under the instructions of a head of the Civil Government (Mayor, Prime Minister etc).
I imagine we are in a MACC situation and not a MACP. If it's all different now, I'm sure someone will enlighten me!