This should never happen. The moment an employee starts thinking about not being honest or transparent about stuff, that may affect their availability or productivity at work (however slightly) is the moment your company has failed. Yes, capitalism will probably keep it afloat for sometime, but personally, I would recommend it to crumble as soon as possible, so that other (better) companies would take the place.
Work relationship is still a relationship. You can't build a meaningful and elevating one on lies and dishonesty. Even if the relationship will be short-lived, both sides need to be honest with each other. "I like that", "I do not like that". When you communicate these things both parties can get an idea of what their partner needs and whether they can adjust to fulfill those needs or not. If there are too many of those "not" - separate and move on.
I have always been vocal about things I do not like at work. Sometimes that had negative effects on me. But I won't stop being vocal, because if you do not say something - nothing will change. If you do say something - maybe something will change even if just a little bit. This is also a reason, why I believe every company needs a flow to allow everyone in the company to submit their ideas in a common pool, which is reviewed by product team.
Back to toddlers: when a person is working from home, they can have myriad of things that can affect their productivity, both positively and negatively. But it's not management's business. They should not care what employee is doing in parallel, even if they are running a fulltime daycare with a dozen of toddlers. What they should care is the employee's ability to do their job to an agreed level of quality. That's what people are being paid for, after all.
Employee sharing information, that they have something, that may affect that quality, and that they are willing to adjust to meet the agreed quality should be appreciated, and not taken for granted, let alone taken with comments like "it's always something with you". There is always something with *you*: processes failing, management hiring incompetent or purely unsuited personnel, you not educating your users (or employees) how to use the product, you over-promising, you outright lying in marketing materials... If employee is putting up with all of these (and probably more), you should be able to put up with 1 toddler. Frame it like "maybe that's our future superstar employee".
