NO : this is wrong .Wrong! The job of Fighter Command was to protect the UK from air attack. Period. That was it's sole role and purpose.
The job of FC was to do what the political and military leadership would order to do and this was depending on the military situation .
In May 1940 Dowding refused to send additional FC squadrons to France because he wanted to keep FC intact .
Would it not be so that the use of FC as an air force in being was better ( in the situations we discuss ) than to wast it in a decisive air battle or in an air campaign of attrition?
The Tirpitz never was fighting against the Home Fleet,but that does not mean that it was useless .Would the Bismarck not have been more useful if it did not leave the German harbours ?
A British fighter lost in an air battle over southern England could not participate in an attack on the German invasion fleet .
An intact FC on SL day would tie a big part of the LW ,resulting in more losses for the German invasion fleet .