In the above sentence, you have used hedging five times with 'I am no expert', 'Sort of appears....', 'I am thinking....', 'might be a good....', Wouldn't you think'?
If this is a scenario where you are a layman and speaking to a person who is a health care professional, it might sound okay, but if you are talking to an other layman for assistance, it is better to be slightly more authoritative and simply say, "He seems to be losing blood. It would be wise to call 911."
Again, as you have mentioned in your article on NY Times, it is not so much as to what is said, but fragile egos that are a problem. You can obviously not order someone more senior or a better expert in the subject matter, but when it is something you rightfully deserve, I guess it should be fine to be assertive. It also speaks a lot about the person you are dealing with. Are they objective and fair or do all their decisions come from a place of ego? In fact, you won't need to be overly assertive with someone who is objective and fair.
Personally, when you meet me for the first time I come across as a person who is so very understated and so very polite that I might actually come across as meek. But, that is until you start poking the bear and you get treated to my true persona when I am pushed. I am anything but meek. That's because I think there is a time and place to assert oneself.