People often bang on about "it's not your home" but us SH tenants have rights. There are very specific circumstances in which SH tenants can be evicted and therefore, while we inhabit the home within the rules it IS our home. I'm sorry private renters don't have that security, I really feel for them. But my home is my home. I might not own it on paper but I won't ever break the rules and therefore it's my home for as long as I want it. People can say that's wrong all they like, but that's the tenancy agreement I signed.
When people buy a property, they don't say it's the banks house! But it is until the mortgage is paid off. Don't keep up with the mortgage and yes you can lose your home. Does that mean you don't call it your home in the meantime? Of course not.
SH tenants are given an empty shell of a place and it is up to them to make it a home. You often don't get even so much as a light bulb left behind! If you want it to be more like a private rental then councils/HA's would need to start decorating and furnishing these homes, providing white goods etc, that's what private landlords do. How much would that cost them?
It's always been the way that if you qualify for it, you can get SH. But it's a roof over your head and it's up to you to make it a nice home. In recognition of that, you're given a secure tenancy, to incentivise you to invest in it. If you want to change it so people only have short term tenancies then the whole model of SH would need to change. That's without all the other problems that would come with that, which other posters have already highlighted.
I won't ever buy my house, so eventually someone else will get it. Whoever does get it, will need it I'm sure. So doesn't really make a difference if it's family A, B or C. To be blunt, SH tenants are dying every day! Just like the rest of the population. Their houses are going back into circulation, as mine will at some point. The fact supply isn't keeping up with demand, is not the fault of any individual tenant. Nor is it their problem to "solve".