Many years ago I heard a mathematician speaking about some open problem and he said, "Sure, it's possible that there is a simple solution to the problem using basic techniques that everyone has just missed so far. And if you find that solution, mathematics will pat you on the head and tell you to run off and play.
"Mathematics advances by solving problems using new techniques because those techniques open up new areas of mathematics."
A proof of a long-open conjecture that uses only elementary techniques is typically long and convoluted.
Think of the problem as requiring spending a certain amount of complexity to solve. If you don't spend it on developing a new way of thinking then you spend it on long and tedious calculations that nobody can keep in working memory.
It's similar to how you can write an AI model in Pytorch or you can write down the logic gates that execute on the GPU. The logic gate representation uses only elementary techniques. But nobody wants to read or check it by hand.
"Mathematics advances by solving problems using new techniques because those techniques open up new areas of mathematics."