Chabadshlucha,
I think you’re misunderstanding Rabbi Millers advise.
What does it mean to respect someone?
What does it mean to engage someone?
Rabbi Miller isn’t saying not to act respectfully to them. On the contrary, he tells you to do just that. Not only is he advising you to act respectfully, he’s advising you to do your part to make sure that they genuinely think and believe that your actions to them are respectful in every way.
At the same time, he’s asking that you not join their fraternity.
You are different and you should know it. Not let them know it necessarily, but you should know it.
Now, the problem is that when you engage people on a daily basis their beliefs thoughts and aspirations often seep into your consciousness. As a Torah Jew, the long term effects can be devastating. So what can a Torah Jew do? (say it quickly, it rhymes!)
The answer is to look at them as if they don’t exist. Not to engage them any more than necessary. It doesn’t mean you shouldn’t be polite, on the contrary, its your job to make them feel comfortable. But just that. Going to workplace parties is obviously wrong. Hanging out with them outside of work is also wrong.
You have to know who you are, you have to know what you’re doing is correct, the rest of the world be damned. that’s where the asylum example comes into play.
You have a job to do, you do your job, and you leave.