Looks like it was more configured for dialup internet direct from an Amiga than network internet connection via remote network gateway.
Nothing in what you quote suggest that, and dial-up also requires routing tables, you know.
A little history lesson... once upon a time there was AS225, the official TCP stack from Commodore, which they made to use with the A2060 ARCNET card and the A2065 ethernet card. Not so much dialup. (ARCNET was at the time a standard competing with ethernet) In version 2 of AS225, AS225-r2, the concept of SANA2 was introduced - Standard Amiga Network Architecture 2 - meaning that the TCP stack did not need to support hardware directly, but could use device drivers, such as a2605.device etc. When CBM folded, AS225 was continued as INet-225 by a company called Interworks, to be sold with the Surfer pack provided by Amiga Technologies.
AS225 and INet 225 if I remember correctly uses "socket.library"
AmiTCP, Miami and Roadshow uses "bsdsocket.library"
This difference means that software has to be compiled against either one or the other, and if you look on Aminet, you will find that most software from the 90ies exists in two versions, one for AS225/INet225, and one for AmiTCP (the rest).
I know for a fact that I have used AS225 with A2065 card in A3000.
I know for a fact that I have used INet 225 with ICard PCMCIA ethernet card from IAM on A1200.
I also remember we (university computer club) inquired the university to sponsor a 10 node license for INet 225 at some point, which of course was rejected.

In the end, AmiTCP by became the standard, AmiTCP 3.0 beta 2 (
http://aminet.net/package/comm/net/AmiTCP-bin-30b2), which was free (and AFAIK the basis of TCP stacks in AROS), and the commercial AmiTCP 4 (
http://aminet.net/package/comm/tcp/AmiTCP-demo-40 that often was hacked) by our Finnish friends NSDi, which was later basis for Genesi and others, though with dubious licenses/copyright infringements. AmiTCP made it to version 4.2 (
http://us4.aminet.net/aminet/docs/rview/AmiTCP4.2.txt)
Luckily there was also Holger Kruse, who in addition to making quite a few nifty software such as AmiWin (
http://aminet.net/package/misc/x11/AmiWin222d) (an X11 implementation that totally rocked after the sources of black&white DaggeX (
http://aminet.net/search?query=DaggeX) got lost in one infamous disk crash), made Miami and Miami Deluxe (Miami with support for multiple interface - use your Amiga as a router). He introduced an improved interface standard for Amiga networking hardware, called MNI - Miami Network Interface - which supported "modern things" that SANA2 does not, such as multicast, promiscuous mode (for tcpdump etc) and more. Sadly Miami was such a success that it was hacked and pirated to a point that Holges said bye-bye and left Amiga to persue happiness working with Carl Sassenrath on REBOL.
In addition to TCP/IP, there was also Envoy (
http://de4.aminet.net/docs/rview/Envoy.txt), which would the equivalent of netbios or appletalk for Amiga, an "office" type networking protocol for sharing resources like disk and printers on a LAN. Later Envoy (3 and up) supported running along with a, or even on top of, a TCP stack.
Anyways, long story short (hehe), INet-225 should certainly work with ethernet.
AmTCP 4.1 or later recommened if you want full functionality. A3.X isn't that hot, apparently.
The big difference between 3 and 4 is the licensing model, AmiTCP 3 was GPL, 4 was commercial.