This question is very important and, in essence, it’s correct.
In the Private Aviation A Private Jet is linked to a base and the VIP crew also have a base.
But the two places not necessarily are the same!
The base of the jet
The base of the jet is chosen by convenience.
It can be the place where the aircraft gets its maintenance (if the aircraft management who manage the jet has also a certification for “part 145” – maintenance).
Or it can be the closest airport to where the owner lives.
Furthermore, it can be an airport with hangar facilities where you can put the jet under a roof and not parked outside.
Or… it can be a combination of these things!
What is important is that in Business Aviation when we speak about the “base” we intend the base of the jet.
And this place not necessarily coincide with the place where the crew lives!
Moreover, some operations have a “floating bases” systems.
This means that the jet itself is not recalled to a specific place but flies around the world and basically never stops for long periods of time, unless for maintenance.
This is the base of the VIP crew.
Means that the crew can stay at their home and they simply commute to the jet at the beginning of their rotations and then back home at the end of the duty.
A gateway is an international airport close the the residential address of the crew member.
The Gateway has some requirements.
They must be well connected in terms of destinations, located at 60 or 90 minutes travel distance from the house of the crew member.
Sometimes the Private Jet Operators make a study about the possible gateways.
Flight connections (how many flights per day/week and to which destinations), airlines costs (some airlines are very expensive, some others are less expensive), infrastructures between the airport and the city (metro or train stations inside the terminal) and, ideally, how often the jet lands in there (to avoid any cost).
Then, they create a list of suitable bases and the crew has to sign up to one of them.
It is on the Operator’s expense to move a crew member from his/her gateway to the jet at the beginning of the duty, and back to the same place at the end of the duty.
Milan has two airports: Linate and Malpensa.
Malpensa is not so used as Private Jet destination because Linate is closer to the city center.
Linate is a very common destination for corporate flights.
If the crew is based in Linate, there are high chances that they don’t need a positioning at all, because they will start and finish their duty with the jet in Linate.
When the owner wants to have the crew close by.
To sum up the previous lines: a Business Jet crew can live wherever they want, baring in mind that there getaway must fit the requirements.
But, sometimes, there are some exceptions.
The owner of the jet might ask the crew member to relocate near the base of the aircraft because he wants to have his Private Jet team always available in case of short notice departures.
They might help you out with some relocation expenses or even provide accommodation, but this really depends on each single situation.