After much pleading the social worker did a home visit last Friday and agreed she does not have the capacity to manage her finances but is unwilling to say she lacks the capacity to decide if she should remain in her own home.
It seems that the issue here is whether your Aunt has the mental capacity to decide where to live in relation to her safety. If she is deemed to have mental capacity in relation to this, and doesn't want to go into a care home, then she can't be forced to go. As per the link that Sirena has provided, Deprivation of Liberty Safeguards (Dols) only applies to someone who has lost mental capacity: 'The person is under continuous supervision and control and is not free to leave, and the person lacks capacity to consent to these arrangements.’
So the care home is right in that if your Aunt doesn't want to be in to their home then they have no legal right to keep her there unless a Dols is in place, however the Dols couldn't be put in place until she is deemed to have lost mental capacity. You need to get a mental capacity assessment first before there is any consideration of a Dols, which can’t be put in place until your Aunt is somewhere where she is not free leave (like a hospital or care home).
She can self-fund and we could do deferred payments until the deputyship is through but no care home is willing to take her unless there has been a deprivation of liberty assessment carried out first. The social worker is refusing to do this and says we can do this privately but there is no mechanism for doing this as depriving someone of their liberty is a power of the state only.
I think the confusion is with regards to what the social worker has/hasn't said. They are unwilling to decide on your Aunt's mental capacity (other than in relation to financial matters) and really this should be undertaken by someone who is knowledgeable about the process. As has been suggested earlier in this thread by Tragicuglyducky, you could request an independent assessment of capacity and pay for this privately. GP’s can be reluctant to do this but you could try an Independent Mental Capacity Advocate:
https://www.gov.uk/government/publications/independent-mental-capacity-advocates