Canada inherited the use of Latin coinage titles from Britain, where (apart from the brief interregnum of the Commonwealth during the English Civil War) a Latin inscription has always been used for the monarch's titles. Britain and Canada are the only two Commonwealth Realms to persist in this anachronism. I suspect that in Canada's case, Latin remains on the coinage for the same reason it is still occasionally used on Swiss and Belgian coins: in an officially multi-lingual country such as Canada, Latin is a "neutral" language choice. If they got rid of Latin, they'd be legally bound to replace it with both English and French, and on all but the largest coins, that would make her titles too small to read.
There may be some residual sensitivity to removing the "D.G". Way back in 1911, the new Canadian coinage issued for King George V was struck, with the obverse inscription just "GEORGIVS V REX ET IND IMP." - they omitted the D.G. Well, there was considerable controversy from church authorities, and from religious folks generally, at this sudden and unannounced "godless" coinage. The following year, the obverse of the coins was redesigned to re-include "D.G". And Canada's never been game to remove it from circulating coins ever since.
There is no legal requirement for "D.G" to be on circulating coins. In Canada, under the Royal Canadian Mint Act of 1985, coinage design is determined by the "Governor in Council" - that is, by the designated government minister (currently the Minister of Finance), acting under advice from the Mint and the bureaucracy, writing the design briefs for new coins, and the Governor-General signing off on them. They're allowed to get rid of D.G. whenever they like.
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