—ignoring a word like marron in favor of brun because it's more accessible to English speakers.
That specific example is probably no big deal because (I believe) "brun" is more common in everyday speech anyway. Of course, it's good to know "marron" too but if the aim is for your students to get by colloquially, "marron" isn't a priotity.
But, on the general principle, I share your concern. If you systematically avoid words that don't sound like English, that's a problem.
—using cognates that aren't quite appropriate simply because they're easier for English speakers to understand (such as using politicien instead of politique, or docteur instead of médecin).
Same as above, more or less. Those precise examples don't seem a big problem to me (at least in the case of "docteur"; again, I think it's a bit more common in colloquial speech than "médecin").
But, on the general principle... well, see above.
—what seems to me to be forcing the French syntax/idiom to conform to English (such as saying prendre un cours instead of suivre un cours, or using marcher where aller à pied or se promener might be more appropriate).
Hmm... yeah, I think "suivre un cours" really is better. Regarding "marcher", it would depend on context, but if you say they use it where other phrases would work better, that's probably true.
All in all, I think you usually have to make
some concessions with beginners so as not to overwhelm them with loads of unfamiliar constructions at once, but too much compromise is bad. Maybe the best way to go (when possible) is to start with things that aren't too different from English but are also idiomatic in French (or whatever language you're teaching; I have more experience of these things with Latin) but you still have to introduce more "alien" stuff at some point.