Why don't we use more powerful routers that have better signal reception instead of adding access points to the network?
I'm not sure I really understand the question, but I think you are confused. Why would routers have better signal reception that WAPs?
A router with Wi-Fi capability merely has a WAP added to it.
WAPs may be placed where you wouldn't place a router (e.g., in the ceiling in the middles of a room), moving the WAP closer to the clients. Multiple WAPs can all be on the same LAN, allowing roaming without reauthentication.
So I think it's important here to get the terminology correct, even though it's VERY common that people use the wrong terms. Properly, an access point allows wireless clients to connect to the existing network (which is usually, but not exclusively, a wired ethernet network). This is distinct from a router, which forwards packets from one computer network to another, and often includes functionality such as network address and/or port translation. The confusion arises because many very common devices, especially home "wifi routers" put all of this functionality into a single device.
The "power" of a router can mean a lot of things, and it's not at all clear what's being asked here. At least one important distinction would be whether we're talking about "how good is this system at sending and receiving wireless signals" but another might be "how good is this device at doing packet forwarding?" - these might well be very different things for a given device.
In large deployments, you can have lots of access points that cover a large area (think about a large commercial building) but none of these access points are actually doing the routing or NAT/PAT functionality.
So... all that said, it will really help us answer the question if you describe in better detail what you're asking and use the terminology most appropriate. I briefly looked over the relevant wikipedia pages and they seem to be reasonably well written: