There're many angles to your question which are problematic. Let's dissect them one by one.
How can there be a profession where some person sits at a desk and trades or invests for somebody else?
Why can't there be? Anything goes as long as people are willing to pay them. There're plenty of people making a living off things which others would consider completely useless. For example, take soccer. If you don't care about soccer, you would also never see the point of paying millions of dollars to 22 adults to chase a ball. But others do, and they pay. This allows those footballers to make a living.
If they are so smart and can predict the market so well, would they not long ago have invested their own money repeatedly in order to amass such a fortune that they would never have to sit and have this as a job for somebody else to make money with their supposed trading/investing skills?
You can be pretty damn sure they're already investing their own money. They don't need yours. That's why investing isn't like fixed deposits - you can just take all your money out and they'll carry on as usual.
Instead of asking this question, ask the reverse. Why should they help you? Does it matter to them if you don't know how to invest and lose money as a result? This is an especially pertinent question given the fact that the stock market is a zero sum game, and the more inexperienced investors there are in the market, the easier it is to make money.
Here's a quote from Warren Buffett:
If I was running $1 million today, or $10 million for that matter, I’d be fully invested. Anyone who says that size does not hurt investment performance is selling. The highest rates of return I’ve ever achieved were in the 1950s. I killed the Dow. You ought to see the numbers. But I was investing peanuts then. It’s a huge structural advantage not to have a lot of money. I think I could make you 50% a year on $1 million. No, I know I could. I guarantee that.
The point here is that he can make you 50% a year on $1 million, but he can't do it with $1 billion or $100 billion. The more money you have, the more you move the markets when you invest, and when the markets move, the less you are able to make. Therefore these people are taking on an active handicap by helping you. The more money you give them, the bigger the handicap.
In other words: you need them, they do not need you. Asking this question then is itself quite offensive.
So then back to the question of why they should help you. Try thinking of some answers yourself. I suspect that pretty soon, you'll be forced to the answer "because I will pay you". Once you're there, then things materially change for them. Yes, they are handicapping themselves by investing your money, but you are paying them, and it might still be a net positive for them.
The entire job seems self-contradictory to me, like somebody working hard every day to push a book on how to become rich quickly. It's obvious in that case that they aren't successful themselves, except in the sense that their book scam makes money by tricking the other people who hope to "also" become rich. Nobody who is actually rich and successful would be pushing a book containing the actual "magic trick" to this success, both because they wouldn't have to work (especially not with something non-rewarding like that) and also because it would be in their best interest to not mention this "secret trick" if they have somehow found it out themselves.
Be very careful with this, because there are scams and there are altruistic people. Go back to the question of why they should help you. Did you also consider "because it's altruistic"? There are indeed self-made millionaires who consider it desirable to educate others (and if you've read stories about destitute people who blew their life savings in bad investments, you might be sympathetic). It's not easy, there are people like you who question their motives. How would you react if you tried to help someone and they reacted with "why are you helping me instead of yourself?"
If you don't want their help, walk away, don't question their motives.
Here's also a brief answer for why those who are familiar with investing might still engage someone else to invest for them, and why these others might be willing to invest someone else's money.
It's less labour intensive. To make serious money, one needs to follow the markets closely. Reading the news is necessary (which can easily be >4 times a day), not a luxury. One also must read annual reports, which are often hundreds of pages in dense legal text. Further, not every one of these investigations actually leads to a worthy investment. It's quite conceivable then that a person will just outsource this to someone else, to free up time for them to pursue their own interests.
By investing someone else's money, they are presumably paying you. That means you have a stable income stream. It's possible to have a stable income stream from investments (e.g. via dividends), but a salary is likely worth more.