Finally, after some tweaking of https://github.com/dillo-browser/dillo-plugin-gemini I'm able to view the #Gemini sites in the DIllo, not the blank page 
Looks like, the gemini.filter.dpi file contains some lines, which works well with GNU sed/awk but breaks with the same BSD utils:
1) according to man 7 re_format, the "\s" can't be used to specify the space character class. Instead of it the "[:space:]" should be used.
2) The forward slash inside the awk's group should be escaped.
It is interesting that both escaped and non-escaped versions works the same way with the GNU awk 
3) Added two options for ansi2html to avoid unwanted second header's content drawn on the page. And to have properly formatted and clickable links in the browser.
Diff beetween gemini.filter.dpi from official repository and my, modified for FreeBSD, gemini.filter.dpi. I fixed some Linuxism/GNUism to make this filter works in the FreeBSD (at least).
The Dillo browser with loaded Nytpu's gemlog (gemini://nytpu.com).
All contents on the main page, even the ASCII graphics, looks good, without any rendering problems.