I think why so many people are drawn to it compared to other professions is:
- Medicine can be exhilirating. It is rapidly moving, your learning is lifelong, and the job involves a high level of practicality and hands-on activity. It is rarely monotonous or boring.
- Medicine is usually intriguing: It is really diverse, and the career is heavily associated with problem soliving, application of knowledge to real life situations etc.
- Medicine is often viewed as fulfilling because you interact with and assist a diverse range of people at a face to face and direct level.
Just thought that some of these reasons might help 
Honestly, say whatever you want but I'm 99% sure you want to do medicine because it'll make your parents happy. In Asia, many families are extremely poor and medicine represents an avenue for a good student to break the cycle and have a decent life. That's why so many parents pressure their kids to do medicine and why 90% of high-achieving Asian students want to do medicine.
Firstly, engineering is also lifelong learning, even moreso than medicine because technology becomes obsolete every few years.
On your second point, the entire profession of engineering is based on applying knowledge.
On your third point, I agree with you.
People think doctors are respected and their job is full of cheer and goodwill but in reality, it's nothing like that. My GP told me that unless you have a personal desire to study medicine, completely uninfluenced by your parents/teachers, it will be the worst decision you make in your life. You miss out almost entirely on family life as you are continuously working, the pay is crap for the hours you work and patients are constantly criticising you and the threat of litigation plays on your mind non-stop. But if you genuinely have a desire to help someone, you'd see this as the only life for you. If you think you're going to enjoy this profession and bask in the sunlight of your prestige, you are wrong.
So why do so many people want to do it? As I said, prestige and their inability to tell their parents that they're making their own decisions in life.
Say whatever you like, but at the end of the day, doctors have to put their hands up a man's anus when need-be, is it really a desire of applying problems and lifelong learning that prompts them to choose this?