I'm a mom, as well as a health care provider, and I was an Air Force flight nurse for many years. I really can see this from all angles-- both the biological need to use the bathroom RIGHT NOW, and the need to have all passengers secured for takeoff. I don't know for certain what phase of flight the plane was in "on the tarmac", but I'd bet they were buttoned up, had received clearance to takeoff, and were in line for takeoff when they wanted to get up.
From the point of biological needs, I do understand that when someone needs to "go" they often can't wait very long. I also understand that safety, and having passengers and small children secured, also can't be ignored during critical phases. Just imagine what kind of lawsuits would be "flying" around for passengers injured during critical phases of flight who refused to stay belted in their seats! To say nothing about the terrorism concerns of roving passengers during critical phases. AND-- the pilot can't just "hold in place" EVERY time a passenger wants to get up and use the bathroom. That would be complete and utter chaos for air traffic controllers.
The fact is, this mother should have done all she could to prevent the need to use the bathroom during loading and takeoff. Up to, and including putting her child in a pull up, if she was worried. And if there was an accident anyway, that is just how it goes.
Maybe it sounds cruel and unreasonable to some not to let anyone up whenever they want, but safety regulations really do over ride the request to get up and use the bathroom. And the current security concerns about terrorism mandate that aircrews have some control over ALL passengers during critical phases of time-- loading, unloading, take off, landing, etc.
It's a 3 year old who had a potty accident. It happens. We wish is wouldn't, but that doesn't mean that how the aircrew handled it was wrong. It shouldn't be news, except that this mother had completely unreasonable expectations for her child, as well as for the air travel experience, IMO.
Heck, I'd be in favor of a free Depends dispenser in the security line, for any adults who might need them!
Air travel is not a pleasant experience anymore (particularly with small children), and no one should EVER expect it to be pleasant or convenient. You WILL be inconvenienced and annoyed pretty much every minute of your air travel experience, from the moment you arrive at the airport, till you leave the airport at your destination. You CHOOSE to fly and accept these annoyances and irritations, and outright invasions of your privacy and rights. And you PAY for it. I fly at least 8-12 times a year for work and pleasure, and have flown with small kids for many years. Flying is not like it was 20 years ago-- the whole experience, apart from arriving alive, sucks, IMO. But it's what you have to accept if you choose to fly.
ETA: We also have NO IDEA what else may have been going on in the surrounding area that the flight attendants didn't tell passengers. Often times there is something the pilots are aware of, that passengers are not yet able to be told. There could have been a security situation, another airplane in trouble, etc. There are lots of reasons passengers need to follow the instructions of the flight crew. The general public can't assume they know more than the flight crew about whether or not it's allowable, safe, or advisable to get up whenever they want. Sheesh.