One of the things that I love about Elementary is the way they demystify Sherlock’s deduction process. It’s like, most other Sherlock adaptations treat his detective skills as a purely inborn quality: a natural, inexplicable talent that’s completely unobtainable by anyone else.
But Elementary goes “you know what? I bet Sherlock put a lot of work into developing and honing those skills, and that he could teach those skills to other people”. Sure, there’s definitely a measure of natural talent to it, but it’s also made clear that it’s just as much about training and learning and depending on others when your own knowledge and skills aren’t enough.
In the process, it’s also tacitly acknowledged that Sherlock couldn’t be what he is or do the things he does if he hadn’t been born so privileged. The only reason why he’s had the time and opportunity to train himself up, to gather such a deep and varied well of knowledge and to form such a vast network of experts he can contact for help, is because he’s rich (and white, naturally, since it ties into his generational wealth and the lenience Sherlock is afforded for his criminality).
Without his father’s money, he would have less time to devote to bettering himself as a detective: he would have to worry about earning a paycheck, about keeping himself fed and housed and keeping his living space clean and orderly without the help of a housekeeper.
He would have to worry about the cost of his experiments and equipment and consulting fees and all sorts of other mundane things that would clutter up his mind. He would have to charge for his services, and accept boring cases just to pay the bills.
Without his father’s money, he wouldn’t have his extensive education. He wouldn’t have the Brownstone. He wouldn’t have met Watson.
And I think part of the reason why Sherlock doesn’t accept payment for his work (mostly, with a few notable exceptions made for entities he despises) is because he’s aware of the advantages he’s been afforded in life.