Only if the family has accepted that the US$15,000 fee is nonrefundable and that vetting may fail. Waiting for the employer's answer costs nothing; paying the fee starts a process that cannot be undone.
The nonrefundable step is the one to delay
The published sequence begins with a processing fee that is not returned under any outcome. That makes it the wrong step to take while a cheaper alternative is still open. Ask the employer for a decision by a fixed date and, in the meantime, assemble what both routes will need: proof of the engineering degree and experience, a clean immigration history and passports for everyone who will move.
If the employer declines, the family can pay the fee and enter vetting with the documents ready. If the employer agrees, the family can still hold Gold Card in reserve. Do not assume that paying earlier secures a place, a rate or a date; the official page does not describe reservations, and its terms may change before the gift stage.
Whatever route is chosen, residence arrives only after the EB-1 or EB-2 process and visa availability, not on the day money is paid.