It could, in principle, but there are several “ifs” and “buts” attached to that.
First of all, notice that I did say a provincial VAT, or value-added tax, but I did not say a provincial GST, because one of the alternatives to contemplate here is that a province would not have to follow the GST base precisely. As long as they are roughly compatible, however, the province could quite reasonably look to the Canada Revenue Agency to collect the tax without incurring significant costs.
As to rates, yes, the base would in practice be broader. You'd expect it to be broader, in particular because you've exposed consumers to value-added taxes on several services. However, at the same time, you're lifting taxes that would otherwise be collected from business inputs, and in Ontario, it's quite a significant number. So you give up a fair bit on that side in revenue by not taxing business inputs. Sadly, it isn't quite as low a rate as you might like to see, but we can certainly contemplate not going all the way, say, to 14%. These are options.