I am not overweight, but I can understand this. I don't entirely
agree with it.
If you want to stick with something objective: if your biggest association with going to see a doctor is being made to feel like crap about your weight, then you will see your doctor less.
I already avoid going to the doctor more than I should, and that's mostly based on inconvenience. If it were a miserable experience, then I know I would procrastinate even more aggressively.
But setting that aside, I think there are a number of reasons why this could make some amount of sense. Doctors do over-focus on weight, not bothering to look for any specific causes because they've supposedly already figured out the problem. Which means the patient leaves with an instruction to lose weight, they don't, and nothing changes. And being fat isn't as highly correlated with health outcomes as people—including doctors—assume. There are plenty of healthy fat people and unhealthy thin people.
There are many other legitimate reasons to control how your doctor approaches an issue. You may have tried the common fixes and know that trying slight variations is unlikely to be fruitful. You are an expert on your own body, with massive gaps when it comes to medical knowledge and details. Your doctor is an expert on human bodies in general, with massive gaps when it comes to specific patients' bodies.
I'm not going to argue that the right answer is to not weigh at all. There are a lot of other options, like not telling the patient the weight. And/or skipping the sermon on weight. Bringing up weight only in the context where it matters. Being specific about what aspect of weight is relevant. (Is it really the mass causing mechanical failure, or are we talking about fatty tissue? Endocrine problems?)
In a perfect world, measuring weight wouldn't be an issue because your doctor would be sensitive yet persuasive, and patients wouldn't be hung up on it and would be making rational choices (eg "I'm fat because it's a side effect of doing things that make me happy and this is what works for me right now.") We don't live in a perfect world. We need to do the best in the actual situation, and handing over all power to the doctor does not have that result.