> For a regular work visa, there's no such requirement; you have until the visa expires
That's wrong. The same rules apply. After 3 months immigration can (but is not required to) force you to leave the country. Both for HSP and e.g. specialist visa types.
For a regular work visa the requirement to engage in activities appropriate to your status of residence can be covered by job seeking etc., whereas for HSP (i) there is no way to satisfy it except working for the specified employer.
That's wrong. The same rules apply. After 3 months immigration can (but is not required to) force you to leave the country. Both for HSP and e.g. specialist visa types.