You've strawmanned the entire crux of your OP. That's not a great move. I'll try clear this up a little, from my perspective:

I suppose the super 'woke' types — unimportant

I would say anyone with eyes would agree that there is a 'too far left'. It's not serious to suggest otherwise, so that's fine.

Right wingers say they/we are just as intolerant because we don't accept their intolerance, lol — unimportant

No, they don't. They only say the first part. The second part is contingent on several things; most of all, whether what's being discussed is some form of intolerance. It usually isn't, in the 'moral' sense. We are all most intolerant of the world around us. Most things aren't what we want to be doing, or choose to be doing and we make great efforts to ensure our intolerance is maintained by not coming into contact with things we wont tolerate. I think that is uncontroversial, if a little under-observed. Now, something interesting is to figure out when "intolerance" becomes 'problematic'. Generally speaking, that's when human rights are being violated - but then, many human rights are also contingent - some (including hte UN i believe) consider internet access a human right. But taking the internet from your misbehaving child is not a form of human rights abuse, in the vast majority of minds.

My experience with the majority of right-wingers i've ever had an actual conversation with is that the things they don't tolerate are generally the aggressive, uncharitable behaviours of others. This is absolutely laden with access for bigots, granted. But absolutely so is the opposite line of trying you best to accept the aggressive, uncharitable behaviours of others as leftists like to do (there's a huge amount of social currency to getting a 'dunk' on the left although I acknowledge we have to be talking about the 'leftist' contingent and not just 'those on the left'. I'm in that camp and I find leftist thinking abhorrent). I think the issue is that in conversation right wingers don't frame their "intolerance" as reactionary - leftists do, which gives it an air of legitimacy on its face that might not be warranted - equally, the disparaging of general right-wing thinking is probably also unwarranted as it usually doens't speak to bigotries, but policy considerations.

Poisoning the well, refusal to engage and immediate labelling of views with words that justify aggresion or violence is rife on the left. These, to me, speak to a pretty intense intolerance - sometimes, of their own. There is some loose empirical data on this.

The upsurge of leftist political violence in the last two years or so seems to suggest that the left is more likely to resort to violence, albeit this is a very recent development as compared to right-wing violence. THe problem is this reflects the same disaparity as IPV does: Women (left) are more likely to engage in violence - but right (men) are more likely to kill more people per event. But stand-alone assassination attempts or successes appear to be a left-wing phenomenon.

If people could just stop for a moment, lay out their goals before speaking to their opponents, things would go much better. My experience is that the right will do this - and be respectful - where the left will not. And are usually objectively wrong about how they've characterised the point they're objecting to.

I suppose the question is what you are intolerant of, not whether you are intolerant. — unimportant

Once one sets out what they mean by "intolerance" and what counts as "more intolerant," the question becomes answerable. For example, if we take "intolerant of X" to mean "does not allow X," and we measure relative intolerance quantitatively, then we merely need to count up the different things that each group is intolerant of. Of course a quantitative analysis will probably be insufficient, but you get the idea.

A core problem on the left is actually an equivocation where they want "tolerance" to mean "acceptance." Once one recognizes that tolerance does not mean acceptance, and that tolerance implies dislike or aversion, much of the muddle coming from the left dries up. The critique from the right is basically a request that the person on the left actually survey the things they are intolerant of, instead of pretending that they are "tolerant" of everything and that it is merely a matter of the tolerant vs. the intolerant.

Tolerance is a term used in many different contexts ranging from what is permitted in intimate situations to legislation that has the power to limit one's freedom. Debate about what is permissible in the latter sense concerns constitutions and the limits to state power. The right to privacy and the establishment of religion is in tension with the demand for equality under the law in the U.S.

The idea of a right wing versus a left wing is different if the aim is to deconstruct the institutions that permit that dialogue to continue. There is that great scene in Vasily Grossman's book, Life and Fate where the Nazi interrogator tells the Old Bolshevik they are spiritual brothers in wanting to rewrite the language of the world.

Amongst the charges made as to who is the real nihilist, this distinction between the worldviews is important. To ignore it is to sleepwalk into history, to borrow a phrase.

Nice.

I wonder what qualifies as a moderate leftist. Here in Australia someone like Biden would be seen as a centrist or possibly a conservative.

I have conservative (not right wing) friends who are certainly more forgiving of people's foibles and differences than some of my left friends, who seem to reach for morality every time they disagree with someone.

Tolerance is a terrible word. To be 'tolerated' sounds judgemental.

Quentin Crisp said something I have often agreed with. Tolerance does not create acceptance. Boredom does. When people experience and are exposed to certain lifestyles or people again and again, the fear or resentment often lifts and what was formerly perceived as divergent just becomes another shade of grey in our lives. I suspect this will happen with trans issues over time.

I keep seeing "right winger" as "right whinger".

There's something very deep in that.

It is like saying 'you are intolerant because you do not tolerate racism' — unimportant

Yep.

Tolerance is often insufficient. It will not do to simply tolerate divergence while still despising it. The further step is to accept divergence. We accept multiculturalism, LGBQTI+, disability and so on as aspects of human variation. Racism, we don't accept, but tolerate; that is, we refrain from denying them civil rights or using coercion against them so long as they abide by the law. This is quite different from accepting racism itself. Acceptance applies to people’s identities, capacities, and ways of life; tolerance applies, in limited fashion, to people whose doctrines we reject.

The grammar of tolerance and acceptance is context-dependent. Acceptance cannot apply to doctrines that deny the very conditions for ethical coexistence. Coherent belief revision requires distinguishing between beliefs about human variation (to accept) and beliefs about harmful ideologies (to interpret but not accept).

The left doesn’t conflate tolerance and acceptance; it simply applies each concept to its appropriate domain. Flattening them into a single “more vs. less tolerant” scale misses the ethical point.

Tolerance is a terrible word. To be 'tolerated' sounds judgemental. — Tom Storm

It is judgmental. A society accepts slavery and then stops doing that.

That is a different cultural war from curtailing expressions of personal identity.