The system you have requires a sensor - so that as you drive, meaning you're in the vehicle - your key is in the ignition and the engine is running.
You have a 2016 - so not sure if it has "PATS" or a simple "Door Ajar" sensor - which is an "upgrade" to notify you - that when you drive your car may have a dangerous condition.
The question becomes do you have PATS or non-PATS.
Why?
Ford started using PATS to help reduce theft and uses the sensors the vehicle can be equipped with, to make it installable.
Which, to accomplish that - means that your key has "passed" the PATS (Passive Anti-Theft System) and will now function as a monitor and alert for the condition you have a potentially dangerous problem on your hands. The hood can be pried up by wind whack and the hood latch could fail and the hood can flip up and damage your car or worse. It's why the latch has a catch to allow you to pop the hood - and then finish the job outside - in a non-moving condition - unless you live in certain high-crime areas where they will find ways to open your hood, as you drive - so the purpose is to keep moving so the target is harder to hit.
When you don't - if and when the event of the hood latch can be pried and lifted - making it appear that someone is trying to get at your car...without you or your key in proximity to it to allow you to do this. The PATS system "arms" the vehicle and sounds the horn in moments like this - where a procedure was not followed. Example - you don't use a pry bar to open the hood (or if you do - then you'd have other issues requiring this method be used) instead you - open Drivers side Car door, pull hood latch and by that "logic steps of action" - the PATS passes the test because the Doors were unlocked, opened one and you pulled the hood latch - so a series of events (more than one type of approach) took place to make PATS "passive" versus aggressive in its behavior - you didn't trigger the alarm.
So, the issue is the PATS system - if the VIN is ok, they don't have to find you a switch, you can short the wires together. but I'm curious about the Hood latch numbers - because the switch is plainly seen in the photo - with the connector - don't know why they'd hassle you on this...
Does the new hood latch have a way to transfer that "cherry switch" to the new one?