Having grown up and am still living among Texas-style conservatives, I can only say this statement is false, and on the border of delusional. I've seen O'Reilly Factor and other conservative news pundits, I've heard Rush Limbaugh and other talk radio hosts, I've been at church and at dinner with conservative table talk. The idea that Republicans are a gentleman's party is simply untrue.
What is true is that certain wings of the Democrat party use demagoguery as a preferred tool, to the detriment of political discourse, and that Trump's almost exclusive use of it has steered conservatives further towards it. But conservatives have never been above demagoguery, insults, or even moral panics, as exemplified by the panics of D&D and rock and roll (and that's not even beginning to dredge up the rhetoric of segregationists).
a) Those people aren't politicians. Bill O'Reilly isn't. Rush Limbaugh isn't. The people around your dinner table aren't. What matters here is the people they support. Think Ted Cruz, he's not throwing slurs around. Think John McCain, he famously scolded people for what they said about Obama.
b) Your experience is very recent. Those sorts of things didn't exist prior to the 90s.
As far as the moral panics go, you are properly identifying it as a conservative thing. That is not the same thing as a Republican thing. There were (and still are in some cases) conservative Democrats. Jack Thompson was a Democrat. I suspect the people afraid Rock n Roll would encourage interracial relations were Democrats. The segregationists you're talking about were Democrats.
Conservative isn't an ideology. It's a pragmatic position. It's just preserving the existing way of doing things. That's different than many political ideologies in that it's totally relative. You have to ask whose way of doing things is being preserved. Republicans and Democrats both had conservative wings in the civil rights era, but Republican conservatives were for the culture of the north, Democratic conservatives were southerners, these two groups weren't fighting for the same things at all.
So you're right, there have always been conservatives who would use demagogery, insults, or moral panic. But that says nothing about Republicans. The difference in the parties is not primarily their views on issues, it's their views on political structures overall. Democrats demagogue because their idea of governance is to represent popular opinion. Republicans see governance as top down leadership and aim to be above the moral common denominator (or at least appear to be so). It's pretty much literally in the names.