Here's the answer to the question: It's true that a router doesn't care about a MAC address nor does a switch care about an IP address.
So why do routers have MAC addresses? Because switches need to communicate with a router and switches only understand a MAC address.
Say a PC needs to send a packet to google.com. First the PC is gonna check it's arp table to see if google is on it's local network. When it discovers that it's not, the switch needs to connect to the router and that can only be done by using the MAC address of the router.
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