Though you are quite right that this is potentially solvable for us: lagrange point sunshields to manage solar incidence would buy us a lot of extra time. But it's still hundreds of millions of years - not billions.
500 million years from now, we're fighting a losing battle against the sun's death.
[1] https://en.wikipedia.org/wiki/Timeline_of_the_far_future
in the next million years (let alone hundreds) humans will have populated most of the galaxy
that is, if we survive climate change
Don't worry, it won't be all. Cats will be doing just fine.
Personally, barring nuclear or bio warfare, I'm quite confident we can figure out how to geoengineer the Earth AND settle into Mars within 500 million years, considering we've come from cavemen to today's technology in only 10000-50000 years.
What worries me is actually after about 3 or 4 billion, where no level of geoengineering can make any of the inner planets of the solar system habitable, and with the red giant sun spewing out chunks of burning crap all over, Jupiter's and Saturn's moons aren't particularly safe either.
Honestly more impressive is us shedding the 4 element theory of nature and the infallible ancient knowledge of Greece in the mid 1600’s (The Sceptical Chymist was published in 1661). We went from a fairly static view of knowledge to seriously considering colonizing mars in 400 or so years. Human growth isn’t linear and we are pretty far to the right on an exponential curve. The thing about sitting on an exponential curve as far to the right as we are is a few steps (a few 100 years) can easily eclipse the total sum of all human progress so far.
In general it would take a very large change in the suns output to make a noticeable difference on earth. There are critical secondary effects and feedback loops etc, but black body radiation is T^4 so you take the square root of the square root of the increase in output. A 5% increase in output should be something like a 1.2% increase in temperature kelvin.
It's a pile of problems related to a common cause (increase solar radiance) that steadily is on a single definite trend.
Put it another way: at 288 Kelvin the partial pressure of water vapor is ~1.7 kPa. At 294K it's ~2.5 kPa. That's a 47% increase (naively). And that line trends upwards only.
To boil the oceans you would need the upper atmosphere to retain that moisture or you just get more weather not net evaporation. 1 inch of rainfall represents a shocking amount of energy flow.
Of course, there are other disasters to come after that one.
Like one, there's a limit on how much water can exist as vapor in our atmosphere, (which is far, far less than what the earth's oceans hold), up until the point of boiling. This is conventionally known known as the saturation point, which is 100% relative humidity. Anytime you try to go above this limit, forces the water to condense and drop down back to the oceans as liquid. So the oceans aren't going to evaporate out to being dry like it predicts unless global temperatures rise by an extreme amount - quite likely requiring boiling temperatures, but in any case, extreme enough that life would likely be dead from the temperature alone. So its subsequent reasoning on how photosynthesis ends is not likely how it will play out.
Additionally, it ignores that CO2 is made by animals, and O2 is made by plants, and therefore, the entire possibility of having a natural carbon cycle without volcanism. Never mind the fact that we currently have climate change tech that produces CO2 already, and if we survive that long, could easily prevent photosynthesis from ending in such a way.
And that's without any human-created CO2.
What do you mean by this? Are you predicting that life on earth can only last for another 1.4 billion years? What extinction event is so certain in the next 400m-800m years?