Your looking at this in a oversimplified way. You got to think deeper then this.
The reality is, the LBGT has a right to work, get healthcare, get social security, be with any sex they want. That was always there, just like for the others.
What those specific republicans (because, yep, not all republicans believed the same) wanted to insure is states rights, as in, to make sure same sex marriege laws for example would not infringe on the right of priests, ministers or pastors to say no to marrying them.
Thats a real issue.
Peoples beliefs, as well as there practices must be free.
And you asked about democrats taking rights? Yea, they wer the party of slavery.
Most of them voted against the civil rights act as well.