The problem is that when government spends $1,000,000 on your halfway house, it provides instant relief. It creates beds. The next year the government does it, the number of employees has doubled and it creates 20% more beds instead of the original amount. The third year it creates no beds and we find that 10% of the beds are gone, and the number of employees is now 1000. The government didn't care how well the money was invested.
Now if you invest $6000 and no one else was doing it, they would probably have created some percentage of 1 bed out of it. And if 18 other people invest $100 each maybe that's enough to complete the bed for a year. And if those altogether 19 people hear that the money went to good use, they donate again and they tell their friends. Maybe the halfway house in 10 years starts earning $25k per year and they keep costs low and the beds start increasing, they rent more space.
The government forced funding breaks it and turns it into a fake jobs program, the community funding it actually makes the service accountable.
This is just bad government, not government.
Public spending on things that require (a) maintainance (as almost all things do) (b) are massively less functional if the investment is not sustained (as many things are) is inherently going to lead to suboptimal results. It's equally bad if private entities do this, but for one reason or another, people seem to bitch about this less, because, well "freedom" etc.
Public spending (private too!) is also bad if it creates unnecessary bureaucracies and unnecessary obstacles. This can be tricky because defining "unnecessary" requires a set of values and these may not be universally accepted.
So, if you have a government program that behaves as in your hypothetical example, then the problem is inefficiency, corruption and waste, not government.