do you have the RPi1 with 256MB or the RPi1 with 512MB?
i am currently flashing Raspbian Stretch to my RPi1 256MB... if it is done, i can tell you if chromium is working there.
EDIT: actually i installed 2017-08-16-raspbian-stretch.img to a 16GB SD card and plugged it into my PRi1 (rev.1) with 256MB.
just without any modification of the configuration, Chromium works more or less.
it is able to download webpages and render them. but extremly slow.
open "
https://duckduckgo.com", page takes ~3 minutes to begin render its content. after total time of 9 minutes, the page was fully loaded and the CPU goes from 100% down to ~15%.
open "
https://google.com", page takes ~3 minutes to begin render its content. after total time of 12 minutes, the page was fully loaded and the CPU goes from 100% down to ~68%.
so, from my side, there is internet working out of the box - chromium is working on a RPi1 rev.1 256MB at 700MHz CPU.
but to me it is not usable, because it is that extremely slow.
(normally i have only the lite version installed on my old RPi1 - that makes more sense to me)
i did not over clock the CPU!
i did not changed the memory split!
i did not update or upgrade anything for that test!
i did not run
raspi-config nor
rc-gui!
i just flashed the SD and boot (and auto resize).
you can over clock the CPU up to 1000MHz, that makes some things working faster, but maybe also a bit more unstable.
you can increase the memory split but on a RPi1 rev.1 with only 256MB, there is not much left when you increase the split.