Yes, proposing that a group have more rights that they don't currently enjoy is just like saying a group should be suppressed and have their rights curtailed.
Your right to drive is unimpaired, but the privilege of driving that you enjoy most definitely is.
Most movements to expand rights to new groups are opposed by those who would thereby experience a diminution of their privelege (which is always relative).
This is also why you have the wealthy so often opposing policies that will create more wealth for everyone including themselves, because those same policies reduce their relative advantages.
I'd love you to show me that there is a repeatable and measurable experiment that determines that as a fundamental function of the universe, people should have equal rights.
Personally, I don't believe that (the fact it's a universal truth) to be true, morals and ethics are a subjective topic and stuff like "everyone should have equal right" is something that most people including me can agree on fairly easily. Other people might not agree.