Why can't routers use MAC addresses instead of private/local IP addresses?

The answer is your software applications and operating system are designed to communicate using TCP/IP, not directly with Ethernet MAC addressing.

So regardless of whether the remote computer/node you wish to communicate with is in your local network or in some other network halfway round the globe, that logical link is established with TCP/IP. That way applications just define the destination IP address in the IP packet, fill it with data (which is actually packed into a TCP/UDP segment), and send it along the way without having to worry how the underlying layers deal with the actual transmission.

It is the operating system's networking stack that determines if the IP address is local or remote:

1. If local network, look for MAC address corresponding to that IP address and send the IP packet direct, which is in an Ethernet frame.
2. If remote network, look for MAC address of default gateway router to send the IP packet, also within an Ethernet frame. The router retrieves the IP packet from the frame, looks at the address and determines the next node (and its MAC address) to pass it along, and sends it that direction in another Ethernet frame.

Note: it is totally possible for IP packets to be delivered in another data link protocol other than Ethernet (guess what, your dial-up or DSL modem is not connected to your ISP via an Ethernet cable). And applications should not have to care so as long as the destination IP address is reachable. It is also totally possible to wrap other networking protocol data (that is not an IP packet) in an Ethernet frame for transmission.

Ultimately, all packets are delivered to a MAC address. When an IP packet arrives at your router and the NAT table shows that it is destined for your private address 192.168.1.10, the router will do a look up to see what the MAC address of 192.168.1.10 is, and send the packet to that MAC address.

However, a MAC address is a layer 2 address, and by definition only allows communication between items on the local network. So the question of "why private addresses rather than MAC address" assumes that all non-public networks are simple single flat networks.

There are almost no private corporate networks that consists of a single LAN, and many domestic environments have multiple networks. In order to route between these networks, you need a layer 3 protocol such as IP.

So a different way to phrase your question would be "Why don't router manufacturers and OS developers create a layer-2-only local network in the specific cases where layer 3 is not absolutely necessary?". And the answer to this question is that it would not be worth creating a niche solution to a scenario where the existing general solution already works perfectly. There would be no benefits whatsoever.