They're two separate markets. Increasing rental supply does drive down rents.
If in one location, rents are very low but house prices are very high, then landlords would have incentive to take properties off the rental market in that location and sell them, perhaps using the proceeds to buy investments elsewhere.
Increasing owned housing supply decreases property costs. Decreasing property costs reduce the costs of renting, and they also make buying a house more competitive than renting, which is why increasing housing supply, not rental supply, drives down rent.
Beyond that, increasing rental supply alone will not decrease rent. It's perfectly viable for rents to increase despite more rental units if the alternative of buying a house got more expensive. See the recent increase in rents as property prices increased, despite no change in rental supply.
And no, pushing up the prices of rents don't incentivize building. Building rentals is meh compared to just buying existing housing supply. While house prices are increasing, so will rental prices commensurately, because renting and buying a house are not isolated and independent choices.
In reality, there is no increase in construction, because the increase in housing demand is no different from someone buying a house, and the increase in property costs that follows it doesn't change anything.