It's a question of grammar - countable and uncountable nouns. The sentence says '..a healthy diet' so it must be followed by a countable noun. You say 'a diet', but it is not normal to say 'a food', especially when you are talking about food in general. It would be grammatically correct to say 'Korean people are famous for having healthy food' (without the article 'a') . However, the phrase 'a healthy diet' is better, because 'a diet' describes the entire balance of food and drink which people regularly eat.

diet is a general name for what people regularly eat.