"The Sun's output can be reasonably precisely computed from the solar "constant" and the Earth-Sun distance at that point. To four places it's quoted as 3.847E+26 J/s."

I suppose having it to four places is accurate enough for this purpose, but the "solar constant" is a averaged number, it seems to me one should use the total solar flux of a specific distance, such as at perihelion or aphelion. Though I suppose that is what you meant.
And computing precisely is misleading if your measurement are not precise enough to warrant it- for example, it seem to me that at four places, the number and areas of sunspots could be part of the measurement.
Anyhow, if you have the reference handy I would like to look at it.

"It was lower in the past due to the core of the Sun being a little bit cooler. As helium "ash" builds up, the core contracts a little in response to the lowered average fusion output per unit volume. The contraction increases the density and thus the fusion rate a smidge bringing things to a new equilibrium - a slightly hotter core with a higher fusion rate."

I don't see why the core in the past would be cooler. Obviously before fusion it's going to be cooler, but before the heat from the fusion reaction first get to the surface, it seems to me that core will be about hot as it gets.

In addition before the fusion reaction occurs in the Sun- you are going to have a hydrogen helium mix- one isn't going to a body of pure hydrogen.
And one is going to have a number factors in addition to the "simple" nuclear reaction, for example it seems mass growth of the star could be a factor. And what about the Sun corona- how did it form, has it changed significantly, and how much energy reaches Earth distance from it.

And there probably many other aspects as to why you have what is called the young sun paradox.

But I wanted to avoid all this, by pointing to much more recent times [less than 100 million years ago] when Earth had an high average temperature and higher levels of CO2 than we have currently.

"Of course Earth's orbital evolution causes what's received from the Sun to vary a bit, thus we get Milankovitch cycles and all that. Antarctica and a largely land surrounded Arctic sea means the planet is susceptible to glaciation cycles controlled by the astronomical cycles too. But the Sun drives the whole show in the long term. Reducing its input by 5% results in freezing conditions unless the greenhouse compensates - which it did half an aeon ago. Increasing the Sun's input by 5% would probably cause the carbon dioxide levels to crash, strangling most of the land plants and driving the biosphere into the remnant oceans. Phytoplankton gets by on much lower CO2 levels and so will survive."

As far as I'm concerned I don't think we have established that 500 million years ago the sun's output was 5% less than the present. I will entertain it as serious possibility. If this is the case, then is it reasonable to assume that 1 billion years ago the sun's output was 10% less. Are we talking about a rate of 5% less per 500 million years, so that the Sun's output has grow at a fairly constant rate of 5% from 4.5 billion years ago? Or do this growth in out put start from say 3 billion years ago?
Give me the theory of how it works.

Meanwhile, I'll give you my hypothesis.
I say oceans warm more than land area and you can't ignore the ocean but rather it's a more significant factor than possible changes in solar output [and/or CO2, Milankovitch cycles, sunspot activity]

So, as I said, take the continents away from the poles and you will have an increase in global temperature by around 5 C. Now, if you remove all the continents, you get even higher global temperature [say, more than 10 C].
So in regard to land masses, it matters where they are, and how they distributed [islands] continent grouping and sizes, etc. It also matters about ocean depth- a large shallow sea could significant affect upon global temperature as compared to a large deep ocean.

Now, in general terms, I would say ocean cause more global warming than land masses. There is no doubt that oceans moderate daily and yearly temperatures. Few doubt that Europe wouldn't be a ice box without the ocean [and to a lesser degree than the hype, the "Gulf Stream"]. Ocean [and water] also cool the planet in terms of clouds and snow- reflecting the sun's energy and as a gas water warms as greenhouse gas. But when speak of ocean causing warmer, I'm not just referring to the greenhouse affect of H2O gas, but the large body of liquid water.

Let's take an extreme example: Mars. Mars receives 60% less solar energy than Earth. With early Earth we talking about receiving 30% less solar energy. So with early Earth we are not talking about anything as bad as Mars today.

Now, what happens if you add 1/2 of Earth ocean to Mars, or say enough water so it's completely covered with ocean.
What happens?
Yes, you are going to get ice. It shouldn't freeze solid, but it would be bright white and reflect sunlight. One aspect is how many millions of years would it remain bright white.
But it's interesting to wonder what Mars would look like after say 100 million years.

But then suppose Mars received only 30% less sunlight as compared to Earth [was closer to the Sun or the Sun increased it's output]. We aren't millions of years later, the ice [actually snow] is bright white. Could parts of the surface of frozen ocean be a warm as the earth's arctic during the summer- could you have open water in some parts of Mars for some times of the year. Once you reach the point of having some open water, you going have the reverse of the runaway affect from snow cover and at some point most of Mars except it's polar regions during the winter will be ice free during the day. You won't have glaciers building up because there isn't any land for it to stack. And if the water is ice instead of snow- it's as transparent as water and doesn't reflect solar energy much more than water does.

Now Earth is different than Mars even if it's receiving the same amount of solar energy. Earth is geological active because of it's moon. And it's geologically warmer simply because it's more massive. Now the geological warmth of earth is utterly insignificant compared to the energy from the sun. But it's also true that solar energy has little to do with affecting geological activity- and not having major volcanic event which spews dust every 1000 years or so on earth isn't likely- so a million or two years of pristine white snow isn't going to occur on earth.
So, instead Mars, if we put Earth at Mars distance- there is no doubt Earth would be cold, but you would probably have some areas of open water and within millions of years the snow would get dirty and melt, and and of course refreeze, so the warmest it might at Mars distance is it might get to Earth conditions during it's coldest periods- which btw, far more hospitable to life than Mars is.
Though if Earth had an additional mile more water than it has at the moment- conditions at Mars distance would warmer than otherwise.

But back to the point, Earth has had periods where had a super continent, and not have a continent in polar regions, it is also true that over time we have more total continental land mass created on earth. So I think these are bigger factors than CO2 content of the atmosphere. CO2 content is a factor but not the main or only factor.