I think there are two things being considered here.
The first would be 'objectivity'
I'm all for objectivity and reason. 'Reason' can be shared but what we think constitutes 'objectivity' is a bit hazy. Math can be pretty objective. That is, one can identify the key axioms upon which reasoning will work and that tends toward conclusions that all participants will share. Science also achieves varying degrees of objectivity. There tend to be a larger number of axiomatic assumptions to be considered in science and many assumptions are more subject to reasonable dissent. Consequently, there tends to be less firm agreement on many conclusions and the results of such 'reasoning' are held more tentatively.
Achieving objectivity in social mechanics and politics is even more difficult. It's not as bad as say, trying to assess art or sport team preferences objectively, but it's very easy to make the case that perfectly reasonable people are likely to disagree about politics or some historical social 'norms'.
Religious objectivity... well. I'm not too sure about that. People within a religious group can agree on any number of axioms or 'divine commands' and reason from them. However, given the less-than-objectively-communicable nature of the starting axioms, you'll find that reasonable people can disagree with the both the starting position and the conclusions.
For example, many axioms Bishop Barron and the Catholic Church consider 'objective' are not shared across other Christian denominations, other religions, or other perfectly reasonable and rational people. Given the set of axioms or 'objective truths' held by Barron, I can reason to many of his conclusions he reaches. The problem is justifying the axioms. Consider something like birth control or the role of women in the church. Many Christian groups diverge significantly from the conclusions of Roman Catholic philosophical tradition.
What I think has happened (and what has happened many times in the past), is that certain axioms -- perhaps bettered described as largely shared assumptions -- are being reconsidered across society. This translates to shifts in behavior and changes in what people conclude. This is not a complete replacement of societal axioms and anarchy, but a shift in a few.
The second issue to consider is 'civility' and the ability to discuss different opinions calmly and respectfully. I would add to this the growing tendency to promulgate and tolerate outright lies.
This is not about 'objectivity' but rhetoric and emotional appeals over reason. Bishop Barren can point to Ms. Griffith, but he could also point to many other examples like the antics of Ann Coulter, Sean Hannity, Laura Ingram, Alex Jones and Rush Limbaugh. And we've seen desperate, despicable acts like the recent shooting of US Congress people on a baseball field, the bombing of abortion clinics, the executions of staff working in the clinics, the Oklahoma City bombing, the Bundy brothers' takeover of the Oregon Wildlife Refuge, the bombing of a recent concert in Manchester UK and the retaliatory car attack on people leaving a UK Mosque. This is not so much a liberal/conservative divide but a ratcheting up of incendiary rhetoric and casting those who disagree as dangerous 'others'.
Still, I'm not in 'The Sky is Falling' camp. Yes, with relatively recent norms things are not so happy, but if we go back 60+ years one can find some pretty horrible characters and events in US history. And frankly, I don't want to go back to the days of even 20 years ago in terms of social norms.