It's fractally wrong and bonkers even by Yud tweet standards.

The worst common electoral system after First Past The Post - possibly even a worse one - is the parliamentary republic

I'll charitably assume based on this he just means proportional representation in general. Specifically he seems to be thinking of a party list type method, but other proportional electoral systems exist and some of them like D'Hondt and various STV methods do involve voting for individuals and not just parties.

with its absurd alliances and frequently falling governments

The alliances are often thought of as a feature, but it's also a valid, if subjective, criticism. Not sure what he means by "frequently falling governments", though. The UK uses FPTP and their PMs seem to resign quite regularly.

A possible amendment is to require 60% approval to replace a Chief Executive; who otherwise serves indefinitely, and appoints their own successor if no 60% majority can be scraped together.

Why 60%? Why not 50% or 70% or two thirds? Approval of whom, the parliament or the population? Would this be approval in the sense of approval voting where you can express approval for multiple candidates or in the sense of the candidate being the voter's first choice à la FPTP? What does the role of a ~~dictator~~ Chief Executive involve? Would it be analogous to something like POTUS, or perhaps PM of the UK or maybe some other country?

The parliament's main job would be legislation, not seizing the spoils of the executive branch of government on a regular basis.

Good news! In most parliamentary republics that is already the main job of the parliament, at least on paper. If you want to start nitpicking the "on paper" part, you might want to elaborate on how your system would prevent this kind of abuse.

Anything like this ever been tried historically?

Yea there's a long historical tradition of states led by an indefinitely serving chief executive, who would pass the office to his chosen successor. A different candidate winning the supermajority approval has typically been seen as the exception rather than the rule under such systems, but notable exceptions to this exist. One in 1776 saw a change of Chief Executive in some British overseas colonies, another one in late 18th century France ended the dynasty of their Chief Executive, and a later one in 1917 had the Russian Chief Executive Nikolai Alexandrovich Romanov lose the office to a firebrand progressive leader.

ChatGPT was incapable of understanding the question.

Now to be fair to ChatGPT, it seems that even the famed genius polymath Eliezer Yudkowsky failed to understand his own question.