The advantage of achieving the goals of equality through legislation or better yet, through a new amendment to the Constitution, is that those means would require more people sharing your goals than not. I come from a state where the State Supreme Court ruled that gay marriage is legal due to equal protection under the law. Regardless of the merits/demerits of that ruling, it caught many people by surprise and caused outrage, and accusations of the Court passing laws instead of interpreting them. Those people will seek to overturn that decision and/or amend the state constitution to their liking for as long as they are able to hold any political power. Though the rights of the LGBT community are the rights of a minority of people, to secure those rights, truly secure them, you will need to win over majority opinion. Look at the Roe vs. Wade decision from 1973. For decades, people, not being able to overturn the decision itself, have attempted to throw roadblocks in the way of abortion through legislation demanding waiting periods or forcing women to get an ultrasound before having an abortion. As a nation, we need to find a way to discuss such issues without vicious name-calling and hate mongering, etc. in order to come to a consensus, one way or another, about controversial issues.
For the record, I'm straight. While my church holds that gay marriage is morally wrong, I believe it has failed to show how this moral wrong affects anyone other than the consenting parties to the degree that it should be against the law. As for other rights, 14th amendment or not, I can't see any legal or moral justification for discrimination with regard to employment or housing for any group of people, so long as they are law-abiding citizens.