As far as I know MAC addresses are never used to route a message to a particular recipient. They're used so that devices that overhear the message, but aren't the intended recipient, can ignore that message on the honor system.

If you have a wired connection, this makes the MAC completely superfluous. The concept is sort of still valid for wireless connections (or of course for "wired" connections where you have multiple devices physically connected by the same wire, a bus, where the concept originated). It should be rethought.

Yes - the whole MAC/ARP/IP parts of the typical IP stack are a legacy mess and should be sorted out.

A typical Ethernet switch operates only at layer 2 and so does indeed route frames by MAC. You need a more expensive, typically managed, switch in order for it to use layer 3 addresses instead and, even then, the initial setup usually relies on layer 2 routing to get an IP address before layer 3 routing can take over. As a sibling comment points out, the whole stack would/does need a rethink to work differently than this. IPv6 SLAAC as mentioned before can potentially eliminate the need for layer 2 addresses but a lot of networks are still IPv4-only, dual-stack, or even layer 3-agnostic.

Note that this is only about last-mile/first-hop. Once you scale past a single LAN segment, routing is mostly layer 3 until you get to core Internet infrastructure which uses ASNs and BGP. At the very least, it is probably sufficient to say that routing across a WAN is all about IP, but if you zoom in on parts of that network of networks, the underlying technologies often use other routing mechanisms internally that don't get exposed to the other parts of the WAN.