Okay. we can set aside the other examples (Mexican rapists, banning a religion Judge Curiel, Khan Family) and just focus on the birther issue exclusively.
The root issue behind the birther position is making Obama illegitimate. It isn't enough to argue against his ideas - he must be silenced entirely by making him disqualified for office. It started with some fringe crackpots who were, shall we say, more overt in their positions. Remember this with McCain?
For some, it's not enough to disagree with Obama. He isn't allowed to be a worthy person with a different position; he must be categorized as completely unfit, unqualified, illegitimate, even dangerous! Now McCain is, at his core, a man of honor. To his credit, his instinct was to shut that batsh!t crazy stuff down. But notice the boos and angst when he says there's nothing to fear from Obama. What is at the root of that fear and hatred? Do you really think its policy difference? It certainly looks and sounds like irrational fear of the "different" guy running for POTUS?
While McCain took an honorable path, others saw opportunity in capitalizing on that irrational fear. Soon a cottage industry popped up to stoke those fears and make those nutters feel like they had a public advocate to help protect them. Remember, the root issue isn't citizenship - its fear of this one "different" man who would be President. The subject of the birth certificate is just a tool selected to take an existing fear/hatred and mask it with something that could seem like a legitimate concern. So instead of coming out and just saying "we are angry about the scary black man with the different sounding name", instead the spin is "we simply want to make sure he isn't defrauding all of us." The loudest and most public of the opportunists to take up this birther argument was: Trump.
Think back to all the previous Presidents who had an organized opposition question their very citizenship. How many did you recall? Zero? Think of relatively recent examples of unpopular Presidents: people from different sides HATED Carter and GW Bush. Their positions were vilified, their character was vilified, but their fundamental right to legally be President wasn't questioned. No, that level of hatred was reserved for Obama.
Obama, the first African American President, is also the first President in the modern history to have an active campaign to question his citizenship. All as a tool to stoke the fires of an irrational fear of this "different" man. One could argue that is just a coincidence (IE: your spitting analogy) - but that would seem like an astounding coincidence - wouldn't it?
Now you might say that this doesn't absolutely prove that 100% of the motivations for birtherism were racism. If you're looking for absolute clarity - most issues couldn't meet the standard of 100% iron-clad proof. However, the preponderance of the evidence is pretty clear. The only real arguments that this isn't racism are (a) People - like some fellow posters - who want to pretend racism and xenophobia don't exist unless they personally see an actual cross burning or lynching, and (b) an academic/semantic argument - like you made.
As for Trump - he is a classic demagogue. I have no idea what he believes - about this or anything else - who could? However, he says racist things, and he has included white nationalists within his strategic inner circle. What is the value in debating whether he actually believes these racist things he says/does, or if he is just using these actions to manipulate the support of people who are racists. Aren't they both pretty negative things?