Obviously things can go wrong, like they did here, but it's not worth discarding the very useful tool of specialized taxes due to cases like this.
Ideally the tax on non-primary homes would be so onerous that people would sell their extra homes to people who do not have homes and that would cause home ownership to increase, taking pressure off of the rental market and driving down prices.
Apartments are supposed to be affordable homes for young adults starting their lives out, not permanent living situations for anyone not financially established as solidly in the middle class or above.
You could easily add waivers for section 8 and military rentals to ensure that all sections of the population would be accounted for, and the tax wouldn't affect apartments and condos, just single homes and duplexes/triplexes/quadplexes that are either being held empty as value stores or used for rental income generators by their owners.
I would also say that once homelessness drops to lower than 1 in 10,000 (a little more than half of it's current number) you could waive 80% of the tax with the caveat that should homelessness increase above that number the full tax would come back immediately. That would provide a strong incentive for property management companies to keep rents low enough for poor people to afford them.