You can have both wired Ethernet and wireless Wi-Fi access on a modern printer (works for me).
The real problem is most users don't understand networking concepts and
assign both printer IP addresses on the same subnet. This causes network routing issues because either your client PC or the router gets confused on which path to send the network packets.
For example:
192.168.1.10 (wired printer)
192.168.1.12 (Wi-Fi printer)
This is why you got:
Reply from 192.168.1.4: Destination host unreachable.
Reply from 192.168.1.4: Destination host unreachable.
Reply from 192.168.1.4: Destination host unreachable.
Reply from 192.168.1.4: Destination host unreachable.
If you have an Ethernet-based network switch or router, and the printer only has the Ethernet port configured, the network knows how to get packets from the PC to the printer.
If you only have the Wi-Fi configured, then your router also knows how to get packets relayed from wired PC's to wireless clients.
When you want to have both at the same time, you can't
assign the printer's two interfaces on the same subnet. Now, you may already have configured your Wi-Fi network to be bridged (sharing the same network as the wired Ethernet). In that case, don't use both interfaces at the same time.
Disable the printer's Wi-Fi port. To have both interfaces at the same time, the Wi-Fi network must not the be the same as the Ethernet's.
For example:
192.168.1.10 (wired)
192.168.2.10 (Wi-Fi)
Either change the Wi-Fi's subnet, or disable the Wi-Fi interface.
