I think that exaggerates the argument into something brittle and thus weaker. Immutability is sufficient but not required.
The issue isn't whether something can technically be changed or not, the issue is whether it is wrong to coerce people in that direction.
Imagine if tomorrow someone invented magic (de)tanning bed where a few weeks of treatments would permanently change your skin color. Would that new capability suddenly make it acceptable for employers to hire only certain shades?
For that matter, we can talk about things like discrimination on the basis of religion, which has already been mutable for all recorded history.
The issue isn't whether something can technically be changed or not, the issue is whether it is wrong to coerce people in that direction.
Imagine if tomorrow someone invented magic (de)tanning bed where a few weeks of treatments would permanently change your skin color. Would that new capability suddenly make it acceptable for employers to hire only certain shades?
For that matter, we can talk about things like discrimination on the basis of religion, which has already been mutable for all recorded history.