I've been delving more deeply into this one and have discovered that the public advice published by the Charity Commission, from which the above quote is derived, is contradicted by their internal guidance for staff. The staff handbook includes the following:-
"4. Constitutional issues for local authorities when acting as charity trustees
In cases where a local authority is sole trustee, it should be remembered that it is responsible in the same way as any other charity trustee for carrying out the normal duties and responsibilities of a charity trustee.
4.1 Who may make decisions
It is up to the local authority to decide, within the scope of local government law, what structures should be used to reach decisions in its name as trustee. An alternative to requiring all the decisions to be reached by the full body of councillors, for example, is to set up a separate committee to discharge its responsibilities as trustees (see section 101 of the Local Government Act 1972).
4.2 Position of individuals acting on behalf of a local authority
Whatever the structure employed, the individuals concerned are not themselves charity trustees. They must, however, act in a responsible way so as to ensure that the local authority acts properly as a charity trustee. If they fail to so this, they may be liable to the council under local government law for any losses it bears as charity trustee (see section 6 below)."
"6. Liability of a local authority and its agents
6.1 The general position in law
The position is similar to that for any corporate trustee (see section 4 above). In principle, a corporate trustee (such as a local authority) is liable for breach of trust in the same way as an individual trustee. It can be sued for breach of trust, and if liable will have to compensate the trust out of its corporate assets.
Whether the individual officers who commit the local authority to an action which results in a breach of trust are in turn liable to compensate that body (in its corporate capacity) is a matter for local government law: it is not our concern. Similarly, any complaints on this topic should be addressed to the District Auditor.
6.2 Indemnity insurance
We should NOT make an Order authorising the purchase of indemnity insurance for the authority as sole trustee out of the charity’s funds. A public body with the capacity to act as charity trustee should be prepared to accept the ordinary measure of legal responsibility for its actions as such, without seeking any indemnity.
Additionally, because any personal liability incurred by the councillors will be to the council rather than directly to the charity, the taking out of trustee indemnity insurance will not be appropriate, because they are not charity trustees. It would not therefore be expedient in the interests of the charity to provide cover for possible liabilities incurred by councillors."
As this appears to relate directly to the subject of your original question, it should be taken as a definitive answer.
Every day's a learning day, as the saying goes!