No, wealth does not accrue to "those who solve people's problems the best".
For example, let's consider what I do, cancer research. In my field, lots of scientists and doctors work very hard to develop new therapies for treating cancer. At the end of the day, this results in a product that is covered by a patent, a form of property.
This property is owned by some very rich people who have never lifted a finger to do any cancer research or solve any problems; all they have done is own things. In this case, because of the specific form of property, they are able to make hundreds of billions in profits without having done any work other than the contribution of some capital. That is, literally, property ownership is the only contribution these people make to drug development, yet they accrue essentially all of the resulting wealth.
The extent of this accrual is a product of the specific forms of property that exist and how much they allow this sort of appropriation.
Every form of property is the product of government - property as we know it cannot exist without government help. For a practical example, until 2013 it was possible to own genes via patents, and about 20% of the human genome was under patent. There were companies that were entirely built on the fact that they owned certain human genes, e.g. Myriad Genetics, which made hundreds of millions of dollars off this. Then the Supreme Court decided this was NOT a form of property, and suddenly this possibility of accrual vanished.
This applies to everything we might think of as property - patents, trademarks, land titles, etc., they exist because of legal force guaranteed by the government.
Some of these property forms are extremely arbitrary measures that seem almost designed to produce wealth transfer (for example, granting mineral rights) to certain individuals.