Engineers/Scientists to break into management consulting Engineers and Scientists are probably not the kind of people management consulting companies look for. But I think many of such would be highly qualified and a good fit for management consulting positions, especially where vision and strategy are needed.
What would you suggest for a scientist or an engineer to succeed in getting a job, considering that the hardest part is to even get a phone call?
I'm afraid I've got to say that many of the things that one might consider an asset, are in reality going to be a hindrance.

Rigour, consistency, reason, domain knowledge, ability to describe uncertainty, always prepared to put appropriate caveats on findings: these are all encumbrances in almost all management consulting.

What you need to cultivate is the ability to form strategy on the basis of a fairly superficial understanding of a business; and the ability to persuade people to pay you lots for you to borrow their watch and then tell them the time.

Oh, and do watch House of Lies through, end to end, a couple of times, to get a feel for it.

Having said that, there is a tiny niche in management consulting available for the skilled technician who's prepared to say no to clients who want "regular" management consulting, and only take on clients who want a thorough, rational, informed job done. There aren't many of those clients, whch is why it's a tiny niche. And it's more likely to be a boutique subject-specific consultancy. But if you find one (or found one, i.e. you are the founder), and it gets the right kind of clients, you'll be very happy (and completely exhausted much of the time), and financially very comfortable while the clients are there.