I agree with David Hammen. Hyperphysics is mostly a very good site but they dropped the ball on that page IMHO. Hope you don't mind a partially speculative answer, but here goes:
Why does it matter if there are some areas of a planet with extreme
temperatures, as long as there are other spots on the planet that are
It shouldn't matter if part of the planet is uninhabitable. There are deserts on Earth which are all but uninhabitable but that doesn't effect life elsewhere. Prior to 5.3 million years ago the Mediterranean sea evaporated and that entire region could have had a salty basin and been hugely hot, but I've not read of life on Earth having any problem with that. Source
Why are "moderate" seasons required for life to exist?
There's a boatload we don't know about the evolution of life on other planets but seems perhaps universally true that life adapts, so I find it difficult to believe that moderate seasons are necessary. Very extreme changes could be difficult, but change can force adaptation.
If humans can live at the equator on Earth where there is the least
amount of tilt, why would an exoplanet with less tilt or no tilt be
The tilt is planet wide but the lowest variation happens near the equator, but animals that thrive near the poles adapt by hibernation or migration and smaller stuff can be frozen and then come back to life, so, I don't agree with the article on this point.
Even if humans could not live on a planet without axial tilt, are
there no other forms of known "advanced" life that can? We know that
extremophiles exist, such as tardigrades' ability to survive in the
vacuum of space. What is the most "advanced life" that could live on a
planet without axial tilt?
One of the interesting historical facts of life on Earth, at least to me, is how long it took what we might consider advanced life to develop. One celled life in various forms was around for over 3 billion years but the first fossils are about 650 million years old. It took life a very long time on earth to get from too small to see to large enough to leave a footprint . . . but, I digress.
I agree 100%, one celled life or Tardegrades could live on a planet with no tilt or 90 degree tilt. Easy. Ocean life in general should be fine cause oceans are more adaptive. Evaporation keeps ocean surfaces colder than land gets during peak heat and while a completely frozen over ocean isn't great for life, cold oceans hold more oxygen and CO2 which can be good for life. Oceans also circulate as an effective means of temperature moderation and fish don't really care how windy it is or how much or little it rains. The tilt question, I think, is really just about life on land.
Land life could be more vulnerable to high wind, extreme temperature shifts, droughts or floods, which could be driven by greater axial tilt, but I find it hard to believe that Axial Tilt is the be-all and end all. Day length and year length are key factors too.
One point I agree with the article on, is that a close to 90 degree tilt might not be ideal with one part of the planet always facing the sun and the other part never facing it but outside of extreme tilts, I don't see why it would be a big deal.
A thick cloud cover, for example, reduces seasonal changes. There's a number of factors.