Countries do make a small (or for some countries, not-so-small) profit from the production and sale of their bullion coins. They would not make them if they could not make them profitably. So in that sense, yes, buying Canadian coins "supports" the Canadian government, and so forth.
Such "support" is higher for countries that actually mine the precious metal in question. Australia, for example, has plenty of gold and silver mines, but no platinum mines - so Australian gold and silver coins are more profitable for the Australian mints (and therefore the Australian governments that own them) than "Australian" platinum coins, because the Australian mint has to import every ounce of platinum from South Africa or Russia.
Finally, some countries receive very little of the profits from the production and sale of "their" coins, because foreign and/or private mints are making them in their name. Fiji silver turtles, for example, are made by the "New Zealand Mint", a private mint (not government owned). The Fijian government hopefully receives a piece of the action, but the lion's share of the profits are going to the private mint's operators.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis