After much stuffing around, I discovered that the A1200's serial port I was using didn't work, so after switching to another Amiga and with lots of patient help from ferix, I have managed to get it working! That's right, I am browsing the web, chatting on IRC, and even directly controlling the PC and seeing its display on my Amiga, connected to the PC via a serial null-modem cable using Multilink. That means I don't even need a TCP/IP stack running, so it's got to be the best solution for low-end Amigas, and the speed is fine! I suppose it's comparable to a fast dialup modem on a good line, which was absolutely fine for the internet until broadband came along.
In fact, since you are using a PC as the host it's possible to browse the web using Opera or Chrome or Firefox through TwinVNC on a separate Intuition screen or in a window on Workbench, so the PC handles the huge websites with its connection and spits out the results to your Amiga quickly.