I looked at the datasheet of a random modern Ethernet chip (ENC28J60) and it's simpler than I expected, doing much less than the old Intel chip, although it includes the low level "PHY" circuitry, which was a separate Intel chip in the olden days. I expect that the newer chip has things like 10-bit counters, but they would be implemented with standard cell logic (i.e. computer-generated layout of gates) rather than the hand-optimized circuitry of the Intel chip.
On the other hand, you can get a chip like the W5300 which includes the whole TCP/IP stack along with ARP and ICMP, presumably running on an internal microcontroller.
https://ww1.microchip.com/downloads/aemDocuments/documents/O...
https://www.wiznet.io/wp-content/uploads/wiznethome/Chip/W53...