Unfortunately, like yourself, I am familiar with being misinformed by my local authority, Herefordshire Council.
I believe your understanding is correct if he is an adult.
As far as I am aware, if he is moving into social care, his care package should be adequate and included things like transport to activities, services and their costs; even outside the area, although they may try to charge a contribution to the cost.
Further, the local authority has a duty to assess for eligibility and needs; however, many councils have policies regarding transport, which can conflict with the law.
I would suggest asking for your local authorities policy regarding social care and transport if they have one.
I have asked my council for policies in the past, when they have claimed to have one in place, only to be informed later they hadn’t actually got the policy they previously claimed to have.
These link should be of help:
The Chronically Sick and Disabled Persons Act 1970
The Care Act 2014 part 1:
Providing information and advice
Promoting diversity and quality in provision of services
I also came across this, but it only mentions in county transport:
Hope this is of use