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Furthermore, the author's claims demote "left" and "right" to some kind of synonyms of what they have labeld as "good" and "bad" on their personal moral compass, robbing them of any systematic meaning in political reality; those definitions match neither scientific consensus, nor the way people self-select into groups that call themselves either "left" or "right", nor the way these terms are used to describe policies. The reason is simple: the goals that he attributes to his idea of what "left" means cannot be achieved and sustained on the scale of modern governments and states without the means of authoritarianism, which is, by his definition, exclusively a "far-right" thing.
Idiots aside, Islamism is obviously far-right. Okay, left- and right- don't have good definitions as is, as they were defined in response to where people physically sat in the Estates General during the French Revolution. But I can at least point to a few properties that are manifestly right-wing and why Islamists all tick those boxes:
1. Right-wing politics supports and defends already-established power structures. Islam (Sunni-Shia divide notwithstanding) is an established power structure; in fact, it's the dominant power structure in the region. Ergo, Islamism supports and defends an established power structure: a specific religion.
2. Right-wing politics is Wilhoitist: it believes in an in-group, to whom the law protects but does not bind; and that the law binds but does not protect the out-group. Islamism specifically wants an Islamic theocracy, that's about as Wilhoitist as it gets.
3. Right-wing politics demonizes the other, to the point of spreading lies about people outside their preferred in-group. Whenever there's someone shitting on Islam you can count on Islamists to make shit up about them to stoke the flames of outrage.
To me, at least, those properties alone are enough to confidently put Islamism in the far-right bin. The fact that some ignorant leftists do the whole "the enemy of my enemy is my friend" bit without actually reading what's on the tin is immaterial.
Once the left is in power, they immediately turn out to be right wing? Does it happen in the very second they gain power or only when try to keep it? Is running for the next election even though you're already in power inherently right-wing?
As for left-wingers becoming right-wingers, yes that can and does happen.
For example, at the time the terms left-wing and right-wing were coined, liberal capitalism was understood to be a radical left-wing attack on the traditional feudal hierarchy of kings and priests that had ruled France in the past. A hundred years later and those same capitalists who had built their business empires on the back of liberal and left-wing revolution were either cozying up to the kings they'd once scared, or establishing new kings to replace the ones that got destroyed.
But merely running for office again does not inherently make you right-wing. Nor does engaging in a coup or other forms of power maintenance make you inherently right-wing. It makes you an authoritarian.