The PIC wasn’t “minimum fuel” on departure from YBHI. The PIC departed YBHI day VFR, no alternate required, with in excess of 30 minutes’ calculated FFR for arrival at YTIB. It was an en-route calculation that indicated the aircraft would arrive with only 29 minutes FFR.
And ATS doesn’t listen to or reply to taxi broadcasts from VFR aircraft at aerodromes in G. Those broadcasts are on CTAF.
I’d be interested in a Centre controller’s view as to what, if anything, s/he would do if s/he heard this broadcast on Area: “Centre and all traffic Broken Hill. VFR Jizzler Alpha Bravo Charlie taxiing for Tibooburra, runway 14, minimum fuel.” I suspect that the reply would be the sound of crickets chirping....