This is probably beating a dead horse at this point, but safe: is definitely the technically correct answer in this case, as the SAFE Network is another way of serving sites rather than another type of domain. It’s most analogous to things like HTTPS, and no-one would ever have considered writing URLs for HTTPS sites as http://website.https instead of https://website.
Besides that, I think that the official SAFE browser should auto-infer the protocol like most normal browsers do these days, so if someone types “base.hdastwb” into their URL bar it should automatically take them to safe://base.hdastwb. This is a much more streamlined way of getting to SAFEsites quickly, and one that would be much more clunky with a TLD than with a proper protocol.
As for the topic of third-party browsers, I agree that Chrome support isn’t something that we should be focusing on, since Chrome is not exactly the best browser for the privacy-aware anyway. However, I would very much like to see a Firefox plugin/distribution/fork for the SAFE Network for some very compelling technical reasons: all of the other browsers (Beaker, Brave, and Chrome) that have been thrown around here are basically based on Chromium (or at least use Chromium’s JavaScript and layout engines) and so they’re all basically the same internally. Firefox is based on different technology, and it has a different set of advantages (such as better support for asm.js and lower memory usage), and I think that having more options rather than one “one-size-fits-all” option is better for users. This is completely irrelevant to the safe: vs. .safenet debate since Firefox actually has a comprehensive plugin interface which supports custom protocols, but I think it’s an important point to make considering some of the other discussion that’s been going on.