My family and I qualify as well and several have been members. I was interested when I was young - the DAR chapters do a lot of very good work - but I grew to have different feelings about hereditary organizations in general.
As for the trans thing, it seems very odd and unlike them. Maybe they’re remembering the Marian Anderson debacle and are overreacting.
The Sons of the American Revolution was founded a year before the DAR--the DAR was founded when the SAR would not let women join.
As far as I know, the issue of women claiming to be men and wanting to join the SAR has not arisen, at least not yet.