Well no, people with capital would be able to buy the houses and then rent it out, extracting the maximum amount they can.
As wages go up, the money available for people to pay for housing goes up.
If a given city employs people and pays enough that 1 person earning an typical wage can afford $1k a month rent, then that's what rent will be. It won't be lower (because landlords will charge the most they can), but it won't be higher (people will move to another city - perhaps one with lower wages, but lower rent).
When we switched to a society where it was normal to have 2 incomes in a given house, the money available to pay for housing increased dramatically (towards $2k, but for families you'd have to remove things like childcare costs which would normally have been done by 1 person), and thus rents increase to the point where supply/demand returns to an equilibrium.
Now aside from rent, you could also buy. A house costs whatever the maintenence is, plus the interest cost of the mortgage (there's also the capital repayment part and the likelihood of the house increasing in value in the future)
Rents are also capped by this - if the cost of buying (including the maintenence) goes below renting, then people will buy. That means house prices are constrained by rental - if the cost of buying/owning house is too much, people will rent. Ultimately they to can always move out of the city and leave their job.
When interest rates went down, people could spent more on buying a house, which meant prices went up.
Imagine a couple that have a budget for $2k a month for housing. If interest rates were to double, mortgage costs would increase. That wouldn't change rents (which would still be $2k), but instead of a house costing say $800k (2k/month allowing 800k loan at 3% interest), instead people could only borrow say 400k. That means that houses go down to 400k, but still cost the same per month.
This is great for people with spare cash, as they can buy up those cheaper properties without having to pay the interest, and still get their $2k/month rent coming in.
It's no change for people who want to buy for $2k/month, that's what they can afford, and it's going to be roughly the same as what it would cost to rent.
It's terrible for people who want to sell and repay their equity, as the house price has suddenly halved. They bought it with an 800k loan, have repaid 10% of that, and owe 720k, now they can only sell for 400k. So they rent it out instead and rent somewhere else.