The template sensor will trigger whenever sensor.date changes state. Then it proceeds to evaluate value_template.
Without entity_id: sensor.date, the template sensor will only be evaluated when Home Assistant starts because, as you pointed out, value_template has no entity and only uses now().
I think that depends on which system of calculation you are using.
And depending on which system the end result you are trying to achieve uses it may make a difference on which system you will need to use.
according to the unix datetime system that you used originally this date (mar 27) actually is in week #12. But according to the isocalendar system used by @123 in their template the same date is in week is #13.
You have to determine which system your target use case also uses or you may be off a week in your template.