I'm not sure I understand this point. You can't expect an economic system to reward you unless you are contributing something that it values. Science is very well rewarded when it produces things like MRI machines, smart phones, vaccines, etc. You can't simply put people with high IQs at the top of an economic pyramid and pay them to do whatever their hearts desire.
I didn't say it should be "required." But if you want more people to know what it is that you do, you start by making them aware of what it is that you do, and explain why it's relevant. In some fields, it's self-evident. But with science the real-world relevance can lag substantially behind the investment of money, time and resources.
Your carpenter example is not a good one. You don't have to get a skilled trade to explain a skill because the skill needs to be performed to given standard (building code for example). If it's not, the consumer has recourse. And beyond that, trades have their work inspected all the time. When I buy a house, I both inspect it thoroughly myself and I hire someone who knows the local code to go through it with a fine tooth comb.
The key point here is that the other fields that require similar levels of training are professions. These are doctors, lawyers, engineers, etc. If you're a doctor it's easy to convince someone or the taxpayers in general to reimburse you for exercising your skill set on you because they understand that it's likely to cure whatever is ailing them. If you're a lawyer you can expect reimbursement because your skill set will help a client to draft a contract that will protect him or her, or navigate a set of problems with very serious consequences. The professions establish colleges that act to ensure those skills meet a certain standard so that the public doesn't need to evaluate individual practitioners.
But science isn't a profession - at least not in that sense. There are no licences or professional standards. It may be embarrassing if you have to retract a journal article, but in most cases no one is going to sue a scientist for making a mistake.