To let simple users see the full hr.employee.form (not the public one) without coding:

1. Go to Settings > Users & Companies > Users
2. Edit the user and assign them to the Employee group (base.group_user) instead of just Portal/Public.
3. This grants access to internal employee views including hr.employee.form.

and solution with code :
A simple way is to inherit the employee form view and make sure it applies to all users, or remove the public view access rule.

Example to force your form for all users (in XML):
<odoo>

  <record id="view_employee_form_inherit" model="ir.ui.view">

    <field name="name">hr.employee.form.override</field>

    <field name="model">hr.employee</field>

    <field name="inherit_id" ref="hr.view_employee_form"/>

    <field name="priority" eval="20"/> 

    <field name="arch" type="xml">

      <!-- your modifications here -->

    </field>

  </record>

</odoo>