I have paid council rates and water (seperately) and STG credited the reward points for it, and that means it counts for qualifying spend.
I have a theory that “government fees” is a legacy clause that is related to the now obselete bank accounts debits and credit taxes, and the banks havent done very well to clarify that in todays context. I think its a very safe bet that if you pay the ATO, you wont get the points (otherwise everyone would be raving about it) but anything else is fine. Unlike some other banks, STG hasnt specifically excluded utilities, insurance, etc.
I just take the position that every bank is different, and every bank can change on a whim, so every time I pay such bills, I always check later if I got the points, and I make a note for my reference. Because I have about 1000 credit cards, and a bad memory, that works for me. So far I havent ever ripped myself off by using the wrong card.