It's ATOC's responsibility to set the eligibility criteria for the Disabled Persons Railcard.
Under s28(3) of the Railways Act 1993, every franchise agreement must include provision requiring the franchise operator to participate in every approved discount fare scheme. For these purposes, s28(5) defines "discount fare scheme" as "any scheme for enabling persons who are young, elderly or disabled to travel by railway at discounted fares, subject to compliance with such conditions (if any) as may be imposed by or under the scheme".
The eligibility criteria are included in the scheme, which is administered by ATOC. The Secretary of State has change control over certain aspects of the scheme, including the eligibility criteria, and therefore wouldn't normally wish to intervene.
Rather than try to assess each person's eligibility itself, ATOC uses a person's eligibility for various Government disability-related benefits and allowances as proxy measures.