Induced demand can cause both to go up, for extra fun times.
You also seem to have missed the "or" presented by my hypothetical low-to-medium-density-city-dweller: either rents and housing prices get higher when you decide to move to my area, or my life changes in ways I don't want as a side effect of your increased supply.
So now let's imagine I own property and can vote, etc. Which of those scenarios do you think this person is going to support? The one that at least gives me a nice valuable house, I'm betting. You still want to move here? Make it worth my while $$$-wise.
I don't think most policy proposals for increasing density have enough obvious upsides for the incumbents to be very realistic.