Kitchen roll- the system that you quote would be unworkable and much more expensive than it is now.
If you are disabled, under the Community Care Act, you can be assessed for your 'needs', recently the calculating of benefits has, had to be taken out of the responsibility of SW.
Most people go to welfare rights organisations. These fill in and go to the appeals for DLA, with the lients. On average 80% of people who have been turned down for benefit, win there case, on appeal. The company that hold the benefit tribunals are staffed by suppossedly experienced, qualified people, but they are getting it wrong. The same would happen under the proposal that you suggest.
Under EU law (so therefore beyond UK law) it is written that benefit decisions must have a right to appeal. The government have a duty to ensure a standard of living for all and have to fight poverty (one again EU law). As others have said DLA is the cheapest way of doing this as otherwise the bill would have to be picked up out of Adult SS and would cost more, anyway. Under various Acts, the law could be used to supply these but that would mean we would have to triple (minimum) the amount of SW's to oversee this. No other professional can assess, under the law, for these services. The welfare rights organisations would head these applications whih would take a well staffed process, to go through. There are time constraints, as there is under claiming DLA, under 'special rules'. Having a lead proffession oversee that would ost the country a fourtune.
Much easier to put this onto families by giving them disability benefits. Look into why the institutions closed, it was the cost that spurred this on.