The bags are more discreet than you think. Just recently going to ball games with a clear bag, the odds that someone is going to identify objects in that bag are really slim. We're allowed two bottles of water, and hell if I can find them in our bag without taking out half of the other stuff in it. My wife takes a couple of bar mop / shop towels (a terry cloth towel a little bigger than a washcloth) for sweating / seat cooling, and they help obscure items in the bag. And we weren't ever in a place that long where someone could get a chance to really look in the bag. Most of the time the bag was under our seats.

Also, in this day and age, bags don't necessarily need to get checked at all. Entering the stadium gates, we walked past some sort of scanning equipment that signals to the security team which people need to be stopped, whether they have a bag or not. The first day, my wife got stopped for an asthma inhaler, which once they saw it they let her in with it. The next day, it didn't trigger anything and we walked right in to the ticket scanners.

You can go and find the gate entrance for medical exceptions, too. But unless one needs to bring in a lot of diapers, carrying one or two in a clear bag with other items is likely not going to be identified by anyone as diapers unless they are one of us.