Yes. The web is about linked HTML documents: if you don't support lynx, links, elinks, w3m, emacs-w3m & eww then you're honestly not doing Web development properly.
You have to do more work to not support them, because writing plain HTML + forms is the baseline.
> Before you know it we'll have someone say "Oh I work for a bank, we have HTML disabled here. Can you make it work without HTML?" give me a break.
That wouldn't be using the Web anymore.
As for those who really are building web applications - well, those are separate beasts. Nobody expects to run Google Docs or Gliffy properly in lynx. But I do expect my banking site to run in a text browser, because there's no valid reason it shouldn't.
(Yes, I understand the business reasons for the current sad state of the web. I also believe those reasons are things one should be ashamed of.)
Support everyone with a browser & internet access, optimize for the stuff that has the greatest market share or where you can do the most good and stand a good chance of having your dependencies met.
Where do you draw the line?
Do I have to also support people who prefer to use command line text-based web browsers?
Before you know it we'll have someone say "Oh I work for a bank, we have HTML disabled here. Can you make it work without HTML?" give me a break.