The site redesign away from the SE "beta" default, including a redesign of the badge icon, was historically something that happened on site graduation from beta status.
We graduated on 16/12/21, as part of a system-wide mass graduation that involved a relaxation of the historical criteria, and which included a total of 59 sites (you might also want to read Open Source is leaving Beta). As we were told at the time of the announcement, "The usual site re-skinning will not accompany our graduation. At some point down the road it is hoped that we'll get a light re-design to move us a little away from the beta skin"; but as the original meta.SE post noted, sites that were in an earlier mass-graduation in 2019 had not, by late 2021, been re-skinned, so I doubt it'll be happening to us any time soon.
That said, if the community here came up with a custom badge icon that passed SE's design police, which had general community support, and which the creator was happy to transfer to SE, it's perfectly possible that The Powers That Be would let us use that, at least until our actual graduation re-design happens. If some candidate emerges, I (and very possibly the other members of your mod team) would happily ask TPTB if we can use that icon on an interim basis. The final decision, of course, will be up to them.
If you're interested in doing that, it might be better to edit this question just to be about the "why", and start a new question for the actual design, to which you submit your design as an answer. That way, the voting mechanism can be a simple indicator of community support for any particular icon.