Misrepresenting 2nd language speakers in fiction

I quite like Hercules Poirot, especially as played by David Suchet. However, I've not read the books so I'm not sure whether the 'Frenchisms' are a faithful reproduction of Agatha Christie's writing or not, but I suspect so. By 'Frenchisms' I mean things like :

Mon ami, what will you? You fix upon me a look of doglike devotion and demand of me a pronouncement a la Sherlock Holmes! Hercule Poirot

 In particular the use of phrases such as 'Mon ami' or even 'Oui'

It's not just Agatha Christie, there's a tendency in a lot of writers to insert words from a foreign language to show foreign-ness of a character. The problem I have with it, from informal observation of others and of my own performance in foreign language, is that the kinds of words and phrases that characters like Poirot are shown using are not the kind of phrases or words that we tend to use in second-languages. Why? Because they are the 'easy phrases' we learn first. The first-language phrases we are likely to carry into our second-language usage are the harder things where we may not know the word. A second-language speaker of English is more likely to say 'my friend' or 'yes' but say something like 'etage one' because they haven't learnt 'first floor' yet, than they are to pepper their English with simple common phrases drawn from their first language.

What we tend to see in these fictional cases is actually the knowledge of the writer of the 2nd language projected onto the foreign-language character, and if the writer is not really well-acquainted with the language in question, what they are likely to know are phrases that a beginner would have learned or (possibly) the stereotypical characterisations of other writers or storytellers. I would recommend that any writer wishing to push beyond stereotypes of speakers-of-English-as-second-language should go to a few classes where people are learning English and pay attention to the way that they are taught common phrases and drilled in frequently-used contstructions and can end up producing sentences which are 'close enough' concatenations of these rote-learnt phrases with vocabulary inserted. Reproducing that kind of thing would have much more verisimilitude as would including misunderstandings where the second-language user responds not quite appropriately because they haven't understood something about what has been said to them and produce an answer or response which is on more familar ground for them.