I don't think it is so much stress, although I do think the cost of these things has become a little ridiculous. It is common manners, no matter what the occasion, that the person whose name is on the invitation, is the person who is invited. Period. It is rude to ask to have someone, whether it is a child or not, invited to a party. If the host had wanted to invite that person he or she would have. ...and nowdays, considering the price of everything 2 or 3 or 10 more people can really add up. Uninvited guests add extra strain on the host.
(off topic, but my mother, in her younger years, used to host the family Thanksgiving. A sit down dinner for anywhere from 18-25 people. Aunts, uncles, cousins, grandparents, etc. My counsin (uncle's daughter) in her teen years began showing up with boyfriend du jour, then she began bringing BF du jour and her mother (uncle is divorced) then BF, mother and maternal grandmother, then BF, mother, maternal grandmother, and sister (from mothers previous marriage) it was always an imposition. There wasn't enough china, or food, or for heaven's sake room at the table!)
As kids get older, it gets easier. They are more firm in who their friends are, and who they want to invite. ...and they have the ability to be more discreet, as not to hurt the feelings of their uninvited classmates. Plus, all my bigger girls want is sleepovers. 5 or 6 girls, some dollar store make-up for "makeovers" pizza, chips, soda and videos. They entertain themseles!