Quoting from the above website,
"When a conjunctive adverb connects two independent clauses in one sentence, it is preceded by a semicolon and followed by a comma."
Without the repeated "he" the second part is not an independent clause. (I must disagree with Jed - we understand the subject is "he" but the second part "then went to do the laundry" is not an independent clause by itself.) Therefore, your reliance on this rule is misplaced.
"He cooked the food, then he went to do the laundry" has two independent clauses. I agree with Jed entirely that in the old days, we would have required either a semi-colon or an "and" to make this grammatical. Many people do write this way. I consider this a comma splice. Others may say the "then" keeps it from being one.
The example sentence on the website of the correct use of a conjunctive adverb is:
"Tuition increases, say officials, are driven by the universities' costs; consequently, tuition income typically covers less than 50% of college budgets."
This has two subjects:
Tuition increases in the first clause and
tuition income in the second clause. The rule applies.
So I'm confused as to why people are saying the example sentence I gave is correct.
What if it were a simpler sentence?
I brushed my teeth and went to bed. -- No problem, right? One subject, two actions.
I brushed my teeth, then went to bed. -- This is the same as your sentence in structure and you've heard all of us say it's fine.
I brushed my teeth; then went to bed. -- This is not grammatical.
I brushed my teeth, then I went to bed. -- I call this a comma splice. Others may not.
I brushed my teeth, and then I went to bed. -- This is grammatical but has unnecessary repetition.
I brushed my teeth and then went to bed. -- This is grammatical and perhaps alleviates your concern.
