The "funny" thing is that up to a couple of years ago when qualification thresholds were lower in continental Europe than in the UK, people were fearing that some people who were really UK resident would become 'Euro cheats' (hate the word) by pretending to be resident in another European country, now that the thresholds are the same everywhere but there are multiple advantages in being registered in the UK (Gold line, 24 hour phone line, BA Amex card with GUF2, transfer of tescos points into avios and more) people might start fearing that people will become 'British cheats' (I will equally hate the word), pretending to be/still be UK residents when they are in fact based abroad!

Also to answer your first question, yes, if you move your account to another country, you won't be able to use the Gold line, silver line, or UK EC service centre in general. Only the call centre in charge of Argentina will be able to book your award tickets, apply your MFUs etc when you cannot do it online.