Each question below assumes that any previous question has been answered with a yes.
Is ille ever used alone as that is used in this sentence?
That is a good idea.
Is illīus ever used alone as of that is in this sentence?
The idea of that alone was worth something.
Is there an ambiguity in this sentence?
Illīus hominis fīlium laudābant omnēs.
I have in mind these two readings:
(a) All praised that man's son.
(b) All praised the son of the man of that.
By (a), Illīus qualifies (determines) and agrees with hominis.
By (b), Illīus refers to an object other than hominis. Maybe we have been referring to a woman as illa, who has married a man, who has a son by a former marriage, who was in general favor.
I am not asking whether any Latin speaker would have used the quoted sentence to mean (b). I am only asking whether (b) could or could not be ruled out as a matter of grammar. If I knew Latin better, I might have tried to come up with a more plausible (natural sounding) example.
What gave me the idea of (b) is the genitive forms of German personal or demonstrative pronouns, such as seiner for er or deren (derer) for die, as in:
Wir wollen seiner gedenken.
Erinnert euch derer, die nicht mehr unter uns sind.
The Latin sentence is from page 124 of Beginner's Latin Book by William C. Collar and M. Grant Daniell (1887).