I've flown with my young children multiple times on both domestic and international flights. We've flown on Alaska, Delta, American, and Qantas. None of them have been smooth and not because of the baby.
Most airlines let you have a child on your lap, without a booked seat, if the child is under the age of 2 during the flight. When flying within the United States this is free, but you still have to add your child’s name to your ticket. When flying international, the above listed carriers require the child have a ticket, which costs 10% of the published fare, but comes with no seat. I don't know the intricacies of this policy; I just know that nobody seems to care about this 10% until you're about to get on the airplane.
Applause to Alaska, the only carrier I’ve flown that lets you add your infant to your ticket via their web site. All the others require you call in and speak to an agent. As a software developer, waiting on hold for 45 minutes, being told by an automated message that I “really should use the website” is especially frustrating. I would LOVE to use the website. Please let me add my infant right on the booking screen instead of as a manual addendum after the fact!
Add in award travel and things get worse. My wife and I recently booked an international flight with Alaska miles, where the second leg was on American Airlines. Our infant was listed on our itinerary from the beginning. At our layover airport, at the gate, 20 minutes before departure, with two squirming babies in our arms, we're told we never paid the 10% infant ticket fee. We could pay it then, plus an additional $45 “airport fee” since we're buying a last minute ticket from a gate agent, or not get on the airplane. Ever since airlines started charging for checked bags I often feel like I'm being nickel and dimed, but this was full out extortion.
Even on a fairly recent domestic Alaska flight, where I was pleasantly surprised I could add my infant to my ticket on their mobile app, while checking in, on my way to the airport, it didn't actually work. I checked the whole family in with the app. Then we checked our bags with an agent, went through security, boarded the plane, and in our seats right before the airplane door closed were asked, "Who's this baby?" Our child was not on the flight manifest and apparently not recorded anywhere in their system. We somehow snuck a baby on an airplane with no questions through 4 checkpoints.
From a process perspective infants flying seems to be common enough that most airline staff have heard of it, but not everyone knows how it works. From a software perspective it is a special case, lower priority, less tested, execution path. And even from the parents perspective the kids are only under 2 for a few flights and then it's normal full fare tickets. So I somewhat understand why ticketing my child never quite works. But still, the hardest part about flying with an infant shouldn't be buying them a ticket.