Personally, I find it a little bit frightening that "ill-informed twitter uses" as you say have the power to sway public opinion, but that fright is dispelled by the simple fact that Twitter, Tumblr, Facebook, Instagram, etc. have become such effective ways to a) communicate with legislators, as they all have their own PR pages, more efficiently than phone lines and letters ever could, b) gather support and centralize public outcry, again, to make communication of issues clearer to legislators, and c) popularize beliefs otherwise seen as radical - i.e. make change. My generation and the one after me (~5+ years after me, I don't know if that's actually another generation or not, but I was born in 1995, so they are all 2000 or later) are both using social media as an outlet to do all of those things. While it might be unfortunate that legislation like this got misconstrued, it's incredibly fortunate that people care so much about issues like discrimination that, even after the Civil Rights Era, they're still willing to make a relatively useless state law in the most useless state in the country a worldwide story. That's effective use of public opinion, and no one ever said that every public opinion has to be right.
Let's focus on Mike Pence, though, as a potential President.
"They refused to do it previously because the law simply didn't need it, and it would have been a confusing and strange clause to have, especially in a state that doesn't have any civil rights protections based on sexual orientation in the first place"
If, in a national debate against the Democratic candidate - presumably Hillary, but I hope not - Mike Pence dropped this line, the response would be "why not?" Then he would have to answer that, and he can't, because I've watched him try before. The fact of the matter is that, if Mike Pence understands politics at all, he doesn't want to have a subject like establishing various sexual orientations as protected classes come up at all. Now that he did this, it will, and he's going to be embarrassed by any remotely moderate opponent he faces on the matter because he's ignorant and incompetent in issues raised by the LGBTQ community, and he always has been. With a plethora of voters, typically those under 30 but nowadays many more whose children have been either exposed to or experienced a gender crisis or simply become involved in issues surrounding LGBTQ people (and any others who are simply well informed and progressive), voting largely on social policy and less so on economic and foreign policy as was more the case in the past two or three four-year election cycles, he's going to lose a lot of support simply by having to answer those questions
That said, I'm sure he'll magically know what to say. He'll try to be somewhat moderate, and he typically is somewhat moderate, and I'm sure a number of people will appreciate an evolved stance on the matter, but he'll still be called a hypocrite and rightly so, not only because of what he's said about this law but because of various things he's said over the course of his governorship.
Regarding the companies, I highly doubt they give a damn about the law. As you said, though, they realize that this law, given the outcry against it, was bad business from the very beginning. When Mike Pence signed it, he was screwing businesses over. If any one of the 20+ major national companies that has threatened to leave Indianapolis does, he'll have screwed the city over, which is why Mayor Ballard specifically came out against the law, saying that Indianapolis was open to all and that discrimination will never be okay in the city. As before, it's not like he's saying something crazy, but he's appealing to the populous in order to keep the city from losing, and Mike Pence should have done the exact same thing.