The 1st Amendment restricts the federal government from passing laws targeting certain forms of speech. If the 1st Amendment is altered to exclude 'hate speech', it is up to the federal government to determine what constitutes hate.
So who gets to decide what is or is not hate speech?
Firstly, Congress can introduce bills defining hate speech. Both chambers of Congress are currently majority Republican, with the majority leaders of each chamber able to determine which bills come up for vote and the majority party capable of whipping up the vote.
Do you want Mitch McConnell, Paul Ryan, and their whips determining what is hate speech?
It falls on the President to sign or veto those laws, as well as instruct and guide the bureaus of the federal government in their implementation. In addition, even without bills, the President can inform bureau priorities and implementations of existing laws via executive order.
Do you want Donald J. Trump telling the bureaus of the federal government what should be considered hate speech?
It falls of the Department of Justice to determine due process and procedure regarding how law is enforced and judged in a court of law. The Department of Justice is currently headed by Jeff Sessions.
Do you want Jeff Sessions telling cops and judges how to determine hate speech in their deliberations of lawful behavior?
Say you sue for your right to speech, claiming it's not hate speech as defined by law by these people, and you argue your way all the way to the Supreme Court.
The Supreme Court is currently majority conservative, with a nontrivial possibility of new court justices seated by President Donald J. Trump. In its current form, Justice Anthony Kennedy and Justice John Roberts are considered the 'swing' votes on the Supreme Court depending on issue. John Roberts tends to be the swing regarding social issues.
Do you want Anthony Kennedy or, more likely, John Roberts ruling on what constitutes hate speech?
If your answer to all of the above is no, then you definitely don't want exceptions made to the first amendment regarding hate speech. Full lstop.
If your answer to any of the above is no, then you likely do not want exceptions made to the first amendment regarding hate speech. See below.
And if your answer to all of the above questions is yes, then you were one of the fascists at the 'rally for the right' demonstration at Charlottesville, and can rest pretty confident that your statements won't be considered hate speech by the current regime.
It's easy to say, "We shouldn't let Nazis march on the streets." If you decompress that pronoun 'we', the folder's contents contain a Senate Majority Leader Mitch McConnell, a House Majority Leader Paul Ryan, a President Donald Trump, a Secretary of Department of Justice Jeff Sessions, and Justices Anthony Kennedy and John Roberts. Those representatives are the ones that get to define 'Nazi.'
And that, my left-of-center friends, is why you don't want exceptions made to the first amendment regarding hate speech.
And don't give me that "Ugh, you're defending hate speech!" bullshit. I'm defending YOUR freedom of speech. You think making hate speech illegal will stop fascists from marching? Charlottesville proves they don't have high regard to following laws. They don't even have regard for human life.
This exertion of violence is assymetrical; the protections of the Constitution are to defend YOU from THEM.