I don't think USB chip is more expensive than Ethernet chip. So, I expect there should exist a board like Arduino, but where USB is removed in favour of Ethernet.
Are there such boards?
It is highly unlikely you will ever find an "Ethernet only" board. There are plenty of boards that have both Ethernet and USB, but one without USB? I doubt it.
To make an Ethernet only board you would first need to create a bootloader with built-in support for the Ethernet port as well as a way of configuring that port to suit your network. Than kind of program would be rather large and take up a fair proportion of the flash memory, leaving you with much less room for your own program.
And as you have rightly noticed, USB chips (especially the dreaded CH340G, the bane of my life) are so cheap now, so why wouldn't you include one on your board?
So you get boards with both USB and Ethernet, and you install the software (generally) through USB which doesn't need any special configuration or complex software to get it to work.
Some boards provide "OTA" (Over The Air) updates, such as the ESP8266, but these generally require the OTA software first installing via the USB (or UART) connection to allow it to communicate on your network and accept OTA firmware. Others, such as the Arduino Yun, use multiple MCUs to provide remote programming. In the case of the Yun you get a "host" microcontroller which is considerably more powerful than the AVR microcontroller that it is programming, which always strikes me as a little pointless.
So no, I doubt that anyone would ever want to make an Ethernet only board.