What conservatives and environmentalists agree on

Liberals see the potatoes, recognize that some people still die even when they eat potatoes like their ancestor, and decide they need more crops.

Like, say, kudzu to enhance the soil and help prevent erosion.

However, unlike the people who actually introduced kudzu, liberals aren't even willing to admit they made a mistake after the fact and will insist that the only reason people object to having their towns and houses completely overgrown with kudzu is irrational kudzuphobia.

Any difference becomes a similarity if we go sufficiently meta. On a sufficiently high level, liberals and conservatives are the same in that they both want to do different things.

But one step below this too-high level, it's -- liberals: "the system is bad, let's change this and this to improve it"; conservatives: "if you tinker with the system, it is likely to fall apart and kill everyone, let's keep it as it is".

Of course, both sides can be expressed by various degrees of sophistication. You can have stupid representatives of both sides, saying things like "the world is perfect as it is, those unsuccessful people just need to stop whining and start working harder" and "hey, let's abolish money and private property, so there will be no more poverty", and you can have educated representatives talking about "progress" or "black swans".

But at the core there seems to be the... feeling(?)... that if you start changing the setting on the social machine for nicer values, the situation will, obviously...
a) ...improve.
b) ...go horribly wrong.

I think it's both. My model is that people join "tribes" that attract them psychologically, because their reflect either their traits or experience, but often also because of peer pressure. And then, the "tribes" create political coalitions to get more power, and this is where many "strange bedfellows" and dogmatism happens. -- In other words, that there are natural "clusters in the opinion-space", and also historical coalitions of clusters based on random events.

The difficult part is to find out, when we talk about a group, how much of its constitution is a natural cluster, and how much is a historically evolved coalition of potentially unrelated clusters.

For example, it is natural for a person to enjoy the idea of a world where their specific traits are highly rewarded, and the skills they miss are considered irrelevant. It is also natural, for people who feel oppressed by a group X, to make "fighting against X" a part of their identity.

But whether groups A and B make a coalition against a coalition of C and D, or whether A and C make a coalition against B and D, that mostly depends on history. Maybe A and B originally had nothing in common, but they joined forces because their common enemy C was too strong at some moment of history; and now it may be different, but the idea that A and B should be allies is already considered common sense between the members of both groups, so C chose D as an ally, despite having nothing else in common.

Talking about "Republicans" and "Democrats" is likely too far on the coalition-making level. Not sure how we operationally define e.g. "conservatives" -- for example, would that include the communists in the former communist countries (people who want to "make Soviet Russia great again")? Because "clinging to the past" seems like a psychological trait, but whether the past happens to be capitalist or communist or islamic or whatever, that's a historical accident.

when the birth control pill and other technology makes new types of social behavior practical ... progressives try to let things develop naturally

Not sure if people who object against sexbots (arguing specifically that they would make women more objectified) are counted as "progressives", because that would be an obvious example against letting things develop naturally when technology makes new types of behavior practical. Similarly, some people who identify as progressives argue against pornography, including cartoons and computer models. -- Maybe this is not a majority opinion among progressives, but still seems like a quite visible one.

I imagine that in some parallel universe, conservatives have legislated mandatory paternity testing for all newborns, and are congratulating themselves for embracing new technology and letting things develop naturally. (The point is, anyone would support things developing naturally in the direction they agree with.)

I think the problem here is that you are trying to argue for a side: the progressives are trying to "to let things develop naturally," and so are behaving in a better way than the conservatives who are trying to "destroy" things.

The truth is this: technology does create new options, as well as taking away some options for some [e.g. many people no longer have a realistic option to "never use a computer"]. This means that technological change drives moral and cultural change.

And yes there tend to be two attitudes both to the technologies and to the behaviors that they allow or prevent. But it is just false that one of those attitudes is right and the other wrong. Rather, in some cases those behaviors are beneficial, and in others they are not. Very often, it will not be clear at first whether the results of the new behavior will be good overall, and only later people figure out that they need still another technology, or they still need to fix remaining problems, or whatever.

Or the weeds of revolution, hippies, and trade unions

I think the conservative attitude towards those things is more likely the environmentalist attitude towards invasive species.

I didn't mean to retract this, but to delete it and move the comment down below.

Historically, Christians objected strongly to fossil evidence that some species had gone extinct. They said God would not have created species and then let them go extinct.

Perfection is a crucial part of Christian ontology. God's creation was perfect. That means, in the Christian way of thinking, it is unchanging. Read Christian descriptions of God (who is perfect), and "unchanging" is always one of the adjectives. "Unchanging" is a necessary attribute of perfection in Christian theology, and God's creation is necessarily perfect. The environment, therefore, was designed and created not to ever change.

One could argue that individuals are thus imperfect because they are born young and then mature. I've never heard a counter-argument against this accusation, though I suspect they exist in the wreckage of medieval theology.