Why does the hint say "Note that with avoir as an auxiliary, past participles never agree with the subject of the verb?
I recently went over the lesson on cases in which the participles do agree, so maybe it should say "almost never"?
Special cases when the past participle agrees (in number & gender) when used with 'avoir' in the compound past in French (Le Passé Composé)
No - in compound tenses, with avoir as auxiliary there is never agreement with the SUBJECT of the verb. As covered in the lessons you linked, including on passé composé, with avoir as auxiliary, there is agreement of the past participle with the DIRECT OBJECT of the verb, ONLY when the direct object precedes the verb in the clause or sentence.
In addition to rechecking the lessons you linked, have a look at the following for further discussion:
For a very entertaining discussion of "why ?" agreement with object before verb and not after, take a look at this great TEDX talk on French spelling - from about the 8 minute mark there is an excellent example of the problem that would arise if there was object agreement when placed after the verb. The talk is in French - but subtitles and great graphics and visuals make it easy enough to follow.
It is quite tricky with avoir to spot when the direct object is appearing before the verb and so can be problematical regarding past participle concordance.
Slightly off topic below --
There is a benefit however when dealing with reflexive verb situations which involves être as auxiliary.
Here, it is very useful to consider just what the reflexive pronoun represents? Direct object or indirect object?
Representation of the direct object will require concordance appearing as it does before the verb (as avoir above) whereas the reflexive pronoun representing the indirect object - no concordance.
So understanding the situation with avoir should be a useful aid when dealing with reflexive situations with être.
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