I think this whole post is predicated on a kind of strawman. Is it really your experience that anyone with socially liberal views will get shouted down as a snowflake or an SJW? If so, I think you've been spending too much time on the internet. It does sound like you are talking about interactions on the internet in your post.
Of course, there are places on the internet where anyone espousing any sort of socially liberal view will get called a snowflake and get told to go back to their safe spaces. Most YouTube comment sections, certain subreddits, etc. Of course, the people doing this name-calling are being intellectually dishonest, but I don't know why you are expecting rigorous and respectful debate in internet echo chambers.
I'd hope you would understand though that this phenonem is not restricted to conservatives, though. Just like I would get called a libtard if I went to a Fox News video and said that it was hate speech, I would get called a monster and probably a racist if I were to go onto LateStageCapitalism and state the common conservative opinion that healthcare is not a right, but a commodity.
In these sorts of echo chambers, no one really wants to understand the opposing side's perspective, so instead of taking a nuanced look at their positions and reasonably judging them, they create some sort of demonization of those positions and judge the other side based on that. In conservative echo chambers, they portrat liberals as hating free speech as babyish and as wanting to take away personal freedoms. In liberal echo chambers, they portray conservatives as either rich people purposefully trying to screw over the poor, or as poor people who are idiotically falling for the trickle-down-economy lies of the rich.
If you want to fairly assess those on the other side of the political aisle, these echo chambers and the extremists that populate them should be ignored, or at least not taken very seriously. I think this is hard to do as the world has generally become more politically polarized in recent years, and nowhere is this more evident than on the internet.
In summary I would say that there is a very small percentage of people that actually believe that all people on the left are babyish liberals who want safe spaces. It is not a problem inherent to conservatism.