No. The described application, vetting, and gift sequence, along with eligibility, admissibility, and visa-availability requirements, apply the same way regardless of how quickly a family wants to move.
Do not let urgency substitute for the actual required steps
Follow the current official instructions in order and avoid treating a family's preferred timeline as though it could shorten vetting or bypass any eligibility requirement.
Keep records of communications and dates for both, and avoid setting a moving date, resignation date or lease based on either process's promotional or general processing language. A programme created by executive action can have its procedures adjusted, its volumes vary, and its published guidance revised, so a timeline described by a third party months ago carries little weight. The case is then processed in the applicable employment-based immigrant category, which involves consular immigrant-visa processing for a family in Canada, medical examinations and visa availability. Payment and even successful vetting do not confer status; the applicant works in the United States only when admitted in a status that permits it, which for permanent residence means after the immigrant visa is issued and used.