It's a problem because of anti-discrimination laws. It's a problem because Christians aren't above the law. Christians shouldn't be allowed to violate anti-discrimination laws while other American citizens are legally forced to obey these laws. It's a problem because LGBT American citizens shouldn't have to deal with religious based prejudice and discrimination just as African-American citizens (and other minority American citizens) shouldn't have to deal with racially based bigotry and discrimination.
If a Christian is allowed to discriminate against a same-sex couple, then that could open the door to more of this:
Mississippi wedding venue refuses to rent to engaged interracial couple because the owners say 'it's against our Christian beliefs to do mixed race and gay weddings' and I don't want to the United States of America begin to revert back to the era before the Civil Rights Movement.
If a Christian feels that he or she can't fully serve homosexuals in their public business because they feel it would violate their conscience (while they willingly serve many other sinners), then they need to find another place of employment and humbly remove themselves from the public eye.
Gay American citizens, as well as other LGBT American citizens, should have the same civil rights and equality as straight American citizens. Gay and other LGBT American citizens should not be denied civil rights and encounter religious based discrimination because some Christian or other person of faith doesn't agree with the sexual lifestyle of that gay person or other LGBT person.
I don't personally agree with the lifestyle of a person who has been divorced and remarried multiple times, but I wouldn't refuse to serve that person or their current spouse if I owned a public business.
I don't personally agree with the lifestyle of a couple who is living together in daily sin before they are married, but I wouldn't refuse to serve that couple if I owned a public business.
I don't personally agree with the lifestyle of someone that I know to be an habitually liar (lying is also an abomination to God), but I wouldn't refuse to serve that person if I owned a public business.
I don't personally agree with the lifestyle of a person who is obese and grossly overweight (gluttony is a sin), but I wouldn't refuse to serve that person if I owned a public business.
As a Christian, I wouldn't single out certain sinners and refuse them service while I willingly served every other kind of sinner under the sun with no moral confliction whatsoever if I owned a public business.