> And just like building bigger highways induces demand so will building more shelters in SF.
"Induced demand" is a fallacy. The demand isn't induced, it was there the whole time and being suppressed by high prices (or, in case of highways, congestion). If there is very high demand, the amount of supply needed to satisfy it is equally high, but that is by no means the same as being infinite and impossible to do.
There are also ways to satisfy the demand for "highways" other than building more highways. Like building more housing closer to where jobs are so people don't have to drive such long distances.
Often these alternatives are better, but sometimes they're not. Sometimes your problem is actually that your highway doesn't have enough lanes. This isn't a question you can answer in the general case without looking at the specifics.
> Not sure why we’d aim to building housing for homeless in the most expensive city in the US
Because that's where a lot of the homeless are because they can't afford the price of housing there. There are literally people in San Francisco who are homeless despite having a full-time job.
Also, it's not "housing for the homeless," it's just housing. Build enough and the price comes down. Then people can afford it instead of being homeless. There is obviously also a benefit to be had for the person who isn't homeless but is spending 60% of their income on an apartment the size of a parking space.