Yes, it might. Especially if you change the word order: Familien mögen die Restaurants. But you would rather use 'diese' to make it more definite, if you wanted to turn the meaning around. The given sentence isn't too clear, but with the not-quite-so-definite 'die' given, I would rather tend to duolingo's translation.
Yes, it's a short-hand way of referring to the restaurant's staff, management or owners.
For example 'the football club wants to buy the player' doesn't literally mean what it says, it's just a short hand way of saying 'the football club's owners want to buy the player'
Common uses you might also see include 'Westminster' used to refer to the U.K parliament, because it's based in the Palace of Westminster and 'the Crown' used to refer to the UK Monarchy.
These are called Synecdoches, which are a type of Metonymn