Suppose that there are some goods that could be provided by City Council, but aren't necessarily best provided by Council. For instance, mowing the lawn on the verge in front of your house. Pretty simple for you to do it yourself while you're mowing everything else, but there are also some efficiencies in just having the Council mower go over it. Some places provide verge-mowing; others leave it to homeowners. If you're a homeowner paying somebody to mow your verge, you pay GST on the service. If the Council mows the verges, you pay for it in your rates; there's GST levied on top of your rates. There's then no tax advantage to Council-provided services over privately provided services.
If instead there were no GST levied on rates, there would be a set of services currently best provided by the private sector that would flip into being Council-provided because of Council's tax advantage.
Petrol levies cover road maintenance and construction, at least in part; the service should be taxed. Excise levies are meant to defray some kind of external cost associated with consumption of excised goods; those costs should also be subject to GST. Sure, we're then heavily over-paying on tobacco, where tobacco excise is multiples of tobacco's cost to the Ministry of Health, but that's a problem with the excise rate, not with the GST. You also don't get the GST back on the amount by which you overpay for something at auction.