Even when we fix a radio with alchemy . . . it takes a life or energy from this world to do it
The explanation, as it is given to the viewer, isn't so much that this world exists for the purpose of alchemy. Rather, the other world is being "tapped into". Alchemists think they are following the principle of Equivalent Exchange; but there is a hidden cost that they are not aware of. This hidden cost is "stolen" from the other world.
Imagine running a cable from your neighbour's house to your house, so you can steal their electricity. Every time you use an electrical device, you steal some of your neighbour's electricity. That's basically what it is for alchemy too.
You put two slices of bread in your toaster. Two pieces of toast come out. That's equivalent exchange, right? To you, living in your house, it may seem that way. But you're simply not aware of the additional cost (the electricity bill) because you're not the one paying for it (your neighbor is paying the cost)
This does not mean that I forced my neighbour to come live next to me so that I could steal his electricity (which is what your question boils down to). Alchemy was simply constructed in a way that it takes the additional cost from another (pre-existing) world.
As far as I'm aware, why the other world exists is never discussed, because then you must also ask yourself why the original (alchemist) world exists, and ponder what existence truly means.