I am not sure I understand your questions. So, I will try to answer them. If you feel my answers aren't adequate, feel free to ask more detailed questions.
No, problems where someone uses a theory isn't enough. They should understand why they use it and know what alternatives exist. They should be able to process this linguistically.
Basically, "The problem is this and I want to solve this problem this way, because this way gives me the resulting information needed."
They should know as much as they can, it should be progressive and taught like we teach languages. Mathematics isn't just a language for solving problems, it can express problems just as easily.
Absolutely, people should be encouraged to try different things. Much like we are encouraged to write an essay, we should encourage people to compose a story with mathematics and it's very much okay to mix it with a second language. Physics is such a thing. In physics, you use both mathematics and your language to expresss and prove. Without both math and a traditional language, some concepts aren't able to be expressed. This is why published papers contain both text and math.
I am not sure about your 'why.' The why is to enable more people to understand the language of mathematics and to enable them to use it to their advantage.
I think that it can be done by increasing exposure and teaching it with greater complexity. It's fine to memorize addition and subtraction tables, but understanding the concepts behind those things is more important.
To be fair, for a short while, we do sort of teach it as a language. This fades out and becomes rote. In early education, we will teach with an abacus and countable objects. We will teach the less than and greater than as a fish that eats the bigger number. Eventually, that stops and concepts aren't considered while memorizing rules is.
I will share a brief story...
I hated mathematics. I absolutely hated it. I didn't understand it. I just did what I was told and gave the answers because I followed instructions.
When I was in sevent grade, I was working on problems which required me to return the area of right triangles. My teacher had stood behind me for some time, I'm not sure how long, before they spoke up.
I remember exactly what they said, to this very day. "You know, all you have to do is square those triangles, find the area, and divide that number in half."
That one instant, my life was permanently altered. At that moment in time, it all clicked. It wasn't easy from there on out. It wasn't a magic moment where I understood everything.
No, it was the moment that I understood that it was a language and that there were many ways to say the same thing. It was that moment when I actually understood that the symbols where actually telling a story. It was that moment when I realized that it was expressing an idea, a concept, and that it was descriptive.
I still stay in touch with that teacher, though they are old and frail. Had it not been for them, my life would be very different. Had it not been for them, for that moment in time, for that effort to make clear, I'd be a very different person.
Most of us can probably have that moment, but little instruction is given that allows for it. We aren't given the chance to see math as a language that is as rich as it is. If we want people to excel at math, we need to find a way to give them that moment.