There is a Canadian joke that is related to this very thing:
Where did the Québécois get that H sound for Ottawa?
They took it out of here.
When speaking with a French Canadian accent, Ottawa sounds like Hottawa, and Here sounds like Ear. There is a similar more adult-oriented joke about the word Happiness. It is utterly baffling to Anglophones, especially since whatever rule they are following doesn't apply to the word Ontario, which pretty much sounds the same as in English.
If you've never heard Quebec French, imagine Celine Dion telling the joke and you'll get it.
The Happiness joke is based both on the missing H and the different emphasis on syllables, so you are probably right or at least close. The subtraction of H in words like Here and Happiness seems like a different rule than its addition on hard vs soft initial-O words, though.
If you've never heard Quebec French, imagine Celine Dion telling the joke and you'll get it.