> I'd love to hear the explanation for how checking up on a toddler every so often is akin to him grossly neglecting his job.

If we generalize that to "what is wrong with frequent distractions" - open plan offices would fall into that category. For an answer, I would direct you to read any one of the many well-articulated anti-Open-plan rants on HN. Toddler care has all the downsides of an open plan office without the (possible) upside of increased team-productivity.

Do you have a toddler? I'm not a parent but my understanding is that you can't expect them to just play quietly by themselves for any more than short stretches.

Depends on the toddler, and on the age.