BTW, my 26 years of good fortune has come from having it serviced by Tag only. It's a bit of a hassle getting into town and it might take a few days but well worth it.
Yes, if you own a Luxury Watch you are always better going to the company who made it for service, a slight word of warning though, if it’s an older model with signs of wear on the dial which holds sentimental value to you, tell them if they’re also replacing the dial tell them you want the old one returning with the watch, the same goes for the bracelet, you will be surprised if you come to sell it how much replacing the dial and bracelet will devalue the watch.
As for the servicing of the movement, especially if it’s an in house movement they should only ever be serviced by the company who made it, if it’s an ETA or Sellita third party movement then yes they can be serviced by most competent Watchmakers, but if they have any complications or modifications added by the watch company they should be serviced themselves for peace of mind, they also guarantee the work they have done.