My experience is that the truly homeless don't beg or panhandle. Most panhandlers don't have their stuff with them. If they have somewhere else to leave their stuff, they're not homeless.
Anyone who honestly needs temporary assistance has plenty of public and private options.
The public options are often dangerous. I wouldn't ever want to be forced to stay there. But you are correct that the truly homeless generally know where they can go to get a bed or a meal.
However, since those options prohibit the possession or use of narcotics, most homeless - more accurately described as bums - do not avail themselves of those options.
There's a lot of paranoia involved in homelessness. They need psychiatric help, but are too paranoid to get it. Also, many have a condition called "anosognosia" where they literally can't see they have a mental illness, so they don't get treatment. It's a much more difficult situation than the usual conservative bumper stickers can address.
It was my understanding that anasognosia was a relatively rare and often temporary condition. Granted though, it is hard to disentangle mental illness from addiction on the street.