Wrong. Long term, subsidies encourage building more housing. Rent control has the opposite effect. On top of that, subsidies can be directed to bring the most benefit to the city (teachers, for example), while rent control benefits whoever happens to be renting at the time it is instituted. There are better long term options, but the political reasons for rent control are because the short term problems can't be ignored, so they aren't actual alternatives to rent control.
Subsidies encourage more housing as much as any other demand-increase does, which is to say, not at all, where supply is effectively constrained. Existing actors in the landlord, real estate, and finance markets all benefit from constrained supply and price inflation.
You've got to break that logjam, and LVT is one of the most effective ways to do that.
Remember: they're not making any more land. You can build out (sprawl, congestion) or up (density). Low land taxes or high improvements taxes both discourage density and encourage sprawl. You cannot change land-use by adjusting demand parameters, long-term and large-scale, only supply and holding costs.
Land value tax requires a constitutional amendment in California. Once again, with feeling: it does not solve short term problems and is therefore not an alternative to rent control.
Also, you've ignored that renters are still encouraged to increase housing supply with rental subsidies. If they have enough votes to enact rent control, they have enough votes to increase housing supply for the long term solution.
Nope. New landlords are incentivized to build housing. The only thing that could stop them is regulations from existing landlords, but in a place with enough votes for rent control or subsidies, existing landlords do not have the votes to stop them.
They might be a reasonable very short-term emergency measure for a few, but long-term, this exacerbates the problem.