You perceived that someone was lacking manners, and you let her know what your thoughts were on the matter. You believe and hope you taught her some kind of lesson by shaming her in front of her children.
When you bragged vented about it here, many of us pointed out that YOU were being rude and lacking manners. Your strident and strenuous defense of your (rude) behavior, amply demonstrates how ineffective it is to go around telling other people that you wish them to conform to your judgment and rule book.
Of course, if she had asked for your opinion, it would be a different story. You, on the other hand, DID post this whole story here to be commented upon. But I guess you only want the attaboys.
Let me ask you this: were the children reasonably quiet, and looking at books or doing other library appropriate things, or were they running amok, yelling, throwing things, tripping people? Then what on earth are you complaining about? Seems to me the mother WAS doing her job by taking the children to an appropriate place for some reading, learning and entertainment.
If a child wishes to sit in a big, comfy chair at a library, and enjoy his book, who are you to stand in the way of that? To a child, placing a backpack on or near a chosen spot is enough to reserve the seat, and only someone with TERRIBLY bad manners would try to take that seat away from him.
If two big comfy chairs are what it takes for two grade-school age children to be enthusiastic about going to a Library, that's one of the best things I've heard all week. IMO, the mother was absolutely right to make sure those chairs were still available for her youngsters when they came back with their books.
Of course, you obviously feel that your right to sit in those particular chairs trumps encouraging young families to spend time in enjoyment and comfort in the Public Library.
Well, you may be right, clearly your own self importance and value outstrips their value. I wonder what would have happened if the child was in possession of a book you wanted? Would you demand that the child relinquish that to you? After all, you are the adult, and by virtue of your age, clearly have more rights than children.