while I previously hadn't done it myself yet, I got the idea because I saw somebody on Reddit mention they used ssh to connect to their
raspberry pi PC running Linux and
viewed websites using something like lynx. The original post has since been deleted unfortunately.
Anyway, because this is Sunday am, and my 7600 was the only classic mac currently a) plugged in, b) connected to the internet, & c) already boots up by default into (supported) pre-X OS I powered that up.
I installed
VNC Viewer Fat, and connected to my MacBook Air (as a proof of concept). And WOW, I do not recommend sending a Full HD screen over 10Base-T Ethernet. I scaled that down immediately. Loaded up iCloud in Safari--apparently High Sierra Safari is too old to access to iCloud. Loaded up iCloud in Edge, logged in. Screenshot. Profit. (Then tried to convert PICT screenshot, and found out the 7600 doesn't even have enough ram for that)
While it's not the fastest, probably 10Base-T Ethernet doesn't help, it did work otherwise perfectly fine! A raspberry pi would just be a smaller package. You could even connect it directly to ethernet if you set up internet sharing just right.
With a raspberry pi you shouldn't have the problem of hidden scroll bars. And I imagine, any browser that can access and handle the full internet will do just fine. You might get more support out of Firefox, but I guess that just depends on the websites.
With XFCE there are quite a few good platinum themes, like
PlatiPlus. All you'd really need to do is make the browser window full screen, probably 800x600 resolution (with website scaling), and hope that Firefox respects your theme scroll bars.