In my area, headscarf wearing woman are probably likely to be Muslim, but might also be members of non-Muslim cultures that also wear headscarves for various traditional, religious-other-than-Muslim, or fashion reasons, so I don’t think it would be that big of a deal if I started wearing one for any reason.
I have worn them while in the Middle East for work, in places/at events where it was expressed that it would be appreciated if women wore the shayla. My biggest take-away was that if I put it on myself (and I know how to, technically), it looks like the Parade of Double Chins, but if someone who really knows what they are doing assists me, the results are in face lift territory, which is pretty sweet.
Did I really answer the question, would I wear one as a show of solidarity? I wouldn’t take it upon myself to do this AT LARGE … but I would if someone asked me to for a specific reason, like if a woman felt uncomfortable being the only woman in a particular venue wearing hijab, or if there was an event at which Muslims explicitly asked for this gesture of solidarity.