The real solution is building more housing. (I think)
Besides, this does hinge on there being enough housing - including a few empty places. It is better to generally have empty units available just like you want a little unemployment to have a pool of employees to choose from.
That's pretty weird, since if HUD assists you with rent, they pay the landlord directly.
At the lower levels, some of our problem is funding by school district instead of distributing state-wide or country-wide. School are still segregated in many ways and the system has created wide disparities between schools because of funding. Sure, you need to adapt a little to the local level, but that's mostly for things like more special education students, more students not speaking english, and/or higher rates of poverty, which affects learning. The last case is some of what has made some of our schools bad, and the rest of us act like we don't care because it isn't our kids.
There is a little merit for the last one, so long as it is reasonable. You want some empty units on the market in most price ranges at all times, just like it is good to have some unemployment so folks can replace workers and have a pool to choose from. You don't really want rental units to sit empty for 6 to 12 months, though, and is probably a sign that there are many short-term rentals or the landlord is charging too much. Even this only works if the landlord can prove he's being prudent with pricing - a simple mortgage + average upkeep + fair set profit percentage would do, so long as it isn't obviously in the landlords favor.
Presumably you'd need some window of exemption for normal transfers and typical moving times.
I'd probably implement as ~2x property tax on dwellings unoccupied for more than 3 months in the past calendar year. As you pointed out, this would have a downward pressure on rents, as landlords would be less able to leave an apartment empty while they wait for someone willing to pay the highest rate. Also, people who own multiple houses would be penalized for keeping some of those houses empty without renting them out.
Every policy helps some people at the expense of others. Concentrated benefits, diffuse costs. You need to weigh both sides, not just the benefits. (edited)
What downstream effects that might have, who knows, but rent control isn't the only solution.