I think the concern being implied here is that a Pentecostal might abuse the power of their high office to cause harm for superstitious reasons.
Kind of pointless to vote the prime minister out of office *after* she's sold the state secrets to China because God promised her it was all part of his plan to bless the country. Or whatever harm it is that the Australian executive can do, bypassing the checks and balancces "for the greater good."
Interestingly, while history abounds with people betraying their community out of a secret loyalty to some higher power or principle, I don't think any of the historical examples are notably Pentecostal. Ed Snowden isn't Pentecostal, as far as I know. Neither is Bradley Manning. Neither was Ronald Reagan, nor is Ollie North. Nor Kim Philby. Nor Margaret Thatcher. Nor Donald Trump.
Closer to home, Paul Keating seems to be greatly disliked. I don't know what he did to piss off so many Australians, but I'm sure many of them are convinced he betrayed his society in some way. Was he motivated by the higher morality of Pentecostalism and a direct "break a few eggs to make this omelette" commandment from his personal Pentecostal deity? Or was his higher morality of some other form?