Sorry for the long post - not aiming to get into a debate. Just hoping share a few thoughts here! I agree with a lot of what you wrote.
Higher interest rates always cool inflation, has happened many times in the past. Inflation is caused by too much demand and money buying too scarce goods. Higher interest rates cool that demand, and prices go down, simple as that.
Just for reference, I majored in economics and am familiar with the concept.

I'm also familiar with stagflation - which is when interest rates do not bring down inflation.
My question was more around the mechanics of the cause-effect of interest rates -> energy demand.
Remember, everything, incl your groceries go up in price when costs go up.
By definition, costs go up when groceries become more expensive. Not sure if this is what you were trying to say - if so, then I agree.
One of the biggest factors there is diesel fuel which has gone up due to higher oil prices caused by lower production not keeping up with demand.
This is a little complicated, but this is my assessment:
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Oil companies lost a huge amount of money in 2020 because covid shut down the economy. The economy shut down led to oil demand plummeting (fewer people driving to work, flying, etc), and OPEC chose not to cut supply. As a result, there was a huge oversupply of oil, and US oil companies lost a ton of money as a result. More than 100 oil & gas companies went bankrupt.
- Now that demand has returned, OPEC has again refused to take action, leading to insufficient oil supply and significantly higher oil prices.
- Wary of taking huge losses again, oil companies are choosing not to make risky investments (example: if profitable at $100+/barrel, and current oil prices are $110/barrel, an oil company will not want to invest in extracting that oil because there's too much risk of it becoming unprofitable in the future - and there are significant costs associated with setting up extraction).
- Lastly, the Russia/Ukraine war cut supply further. However, the US recently made overtures towards normal relations with Venezuela, which may help stabilize gas prices.
And when prices go up, there is pressure on wages, which then causes more inflation.
From what I understand, increasing wages applies pressure on prices to go up. I'm not sure how stable the relationship is between higher costs and higher demand, without strong unions.
Same with house prices and cars. People are buying them because borrowed money is cheap. Higher interest rates will quickly cool that.
Definitely agree here, although I think some other factors are playing into the housing market as well. In particular, covid has led to a surge in tech workers leaving the expensive cities and buying up property in cheaper areas, while working remotely. I'm not sure to what degree this is the case in Canada, but it's definitely true here in the states.
Only problem is that we are late with raising interest rates, because they thought it was "transitory". Now the problem is much worse, and will cause considerable more pain to fix. IE maybe a recession and lost jobs, not too mention a lot of investment nest eggs shrunk badly.
Agreed. Startups in particular are in trouble. Access to funding has evaporated in the past few months, and companies who cannot ride this out will not have any favorable options. Either be acquired cheaply, or shut down, or raise funds with very unfavorable terms.