It's almost LOL wrong to state that conservatives have gladly accepted that trans people are entitled to full and equal treatment under the law.
Conservative groups have sued to get religious exemptions to Bostock (https://news.bloomberglaw.com/daily-labor-report/religious-g...) which is particularly pernicious given the number of religious organizations that operate large employers like hospitals.
Just one Tucker Carlson clip (https://www.foxnews.com/video/6324164359112) shows how American conservatives are continually frothing at the mouth about trans rights. There is a related wave of anti-trans laws being pushed through by conservative elected officials at the state level right now. They're specifically attacking trans children and school policy, as the soft underbelly that gets conservative parents riled up for a culture war and which is legally more vulnerable.