Only Australian and New Zealand Citizens have unrestricted right to work in Australia. All other country citizens require an appropriate Visa.
"Applicants who are nominated by a State or Territory government agency under a State Migration Plan." I believe each State has access to a number of "discretionary visas", possibly 500? You won't gain access to those.
An "Aeroplane Pilot" is ANZSCO Code 231111 and is within the code range to apply for a 457 Visa, ENS or RSMS Permanent Residence, but not included on the Skilled Occupation List (SOL) – Schedule 1.
There is nothing to stop a potential employer offering you emplyment and applying to sponsor a Visa, however it will be incumbent upon that employer to demonstrate that no suitable Australian or New Zealand applicant is available.
And I imagine that would be relatively or extremely difficult unless you had specialised experience or ratings that are in demand and for which there is a critical under supply of labour or absence of applicants in Australia. If the position you seek is a general aviation or general commercial pilot position, I suspect Visa approval wil be impossible.
Also, if the job is city based - Brisbane, Sydney, Melbourne etc - Visa approval will be more difficult to obtain than if the position were rural or regional based.
I think that is the way it works............ Don't take my word for gospel, talk to an Immigration representative at an Australian Embassy or a reputable Migration Agent.