Accent Confusion

I've been thinking about accents. Namely, how to write characters that have them. Writers usually indicate a character's accent in the following ways.

1. Phonetic Approximation
   When authors reproduce the sound of accented words, which leads to French characters saying things like, zee instead of "the" and Scottish characters mispronouncing every word that isn't a Scottishism like "Och!" "Lassie," or "ken".
2. Foreign Phrases
   Authors often sprinkle a character's words with non-English phrases. For example, French characters will say things like, "Mon Dieu! He is a spy. We must inform Le Resistance!" And Russian characters will say "Das Vidanya ." when they leave.
   

But here's the thing. I know or have known people whose first languages are Spanish, French, Hatian Patois, Russian, Georgian, Czech, Chinese, Korean, Thai, Vietnamese and Japanese. None of these people sprinkle their English-language speech with words from their native language. If anything, it's the other way around, as I often hear English phrases mixed into their conversations with family and same-language friends.

Aside from the phonetic differences in pronunciation, In my experience the thing that distinguishes a non-native English speaker is sentence construction, and, sometimes, gendered pronouns. For example, a native Spanish-speaker might say, "I dreamt with..." instead of "I dreamt of..." because "Dreamt with" is a direct translation of the Spanish construction, "Soñar con...".

I much prefer to use word choice and sentence construction when portraying an accent, but the problem is, the reader may be unfamiliar with these cues. Which I guess is why so many authors use phonetic approximation or foreign phrases. What to do?