The full part you're quoting is this:
Persons who may carry out specified work under supervision
6.—(1) This paragraph applies to a person who carries out specified work in a school where—
(a)the person carries out such work in order to assist or support the work of a qualified teacher or a nominated teacher in the school;
(b)the person is subject to the direction and supervision of a qualified teacher or nominated teacher in accordance with arrangements made by the head teacher of the school; and
(c)the head teacher is satisfied that the person has the skills, expertise and experience required to carry out specified work.
(2) In this paragraph a “nominated teacher” means a person who is mentioned in paragraphs 2, 3 or 4 of this Schedule and who is nominated by the head teacher where the head teacher considers that appropriate in the circumstances.
(3) In determining whether the person mentioned in sub-paragraph (1) has the skills, expertise and experience required to carry out specified work in a school, the head teacher may have regard to—
(a)such standards for higher-level teaching assistants, or guidance concerning school support staff as may be published from time to time by the Secretary of State; and
(b)such guidance as to contractual matters relating to school support staff as may be published from time to time by any local authority or other employer.
The crucial difference between planning as a HLTA and a teacher is where the accountability lies.
An HLTA can plan but the teacher remains accountable for the class and the progress. This is why the teacher (either class teacher or appropriate teacher in school) needs to have oversight of what is going on.
(Edit to add - for example there's a difference between planning some lessons to deliver on a specific topic Vs designing the curriculum and deciding without support which topics are being studied next half term)
Good HLTAs are worth their weight in gold and are paid awfully for what they do.
There's also sometimes problems when teachers who are fed up with the pressure of teaching take HLTA roles without fully appreciating exactly what can be asked of an HLTA, then complaining when it turns out the HLTA role is closer to a teaching role than a level 2/3 TA role.