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The problem isn’t lack of shelter. There’s enough shelter available for the homeless. They just choose not to use it because it comes with rules like no drugs and (often) no pets.
No there’s not enough shelter available for the homeless. Shelters have occupancy limits and especially in the US most states do not have enough space. Some states have less than half the beds needed to shelter their states homeless population. https://endhomelessness.org/homelessness-in-america/homelessness-statistics/state-of-homelessness/#homeless-assistance-in-america