This is a good way to think of it, but let me interject some of my own opnions into it.
Math is like a logical framework we can use to model phenomena.
This is indeed true, and should be the key takeaway of the discussion. But math is even more abstract than that.
Math is just a set of ideas. That's it. Those ideas are useful if
- They're consistent
- They model what we see in reality
Those bullets are not requirements, but they're helpful. And if your system is inconsistent, then it's even more useless.
You can say that everything turns into nothing, but such a model is useless because you can't apply it to anything.
There is no truth in math in that you're not going to find evidence that math exists in the universe. It's just a tool that we can use to great benefit.
OP's idea is basically, "Should I really be hammering nails? Nails contain iron, and I need to increase my iron intake, so I should be eating nails, not hammering them."