'They' is a perfectly acceptable singular pronoun, and has been in use that way for centuries. The idea that 'they' isn't a singular pronoun comes from the same stable of grammar rules that say it's not okay to use a preposition to end a sentence with.
Although I agree that you can end a sentence with a preposition if you want to, your sentence is still grammatically wrong. What you wrote is equivalent to "...it's not okay with which to use a preposition to end a sentence." Maybe try "a preposition is not an okay thing to end a sentence with."
Edit: My version is equivalent to "a preposition is not an okay thing with which to end a sentence."
Honest question: I often find I refer back to the rules of grammar when they prevent ambiguity or enhance clearly (often two sides of the same coin). With this phrase, what ambiguity is prevented or clarity enhanced?
It's a mechanical transformation. "End a sentence with" <-> "with which to end a sentence". As I pointed out above, vacri's version turns into an obvious jumble after this equivalent transformation, which means the original is also a jumble. It's just a more difficult jumble to untangle, so you might not notice it's wrong.
The relevant concepts here are preposition stranding and pied-piping. These are not "mechanical transformations" in the sense you mean -- there are certain constraints on when a preposition can be pied-piped and when it can't.
Maybe there's a better way to show it. But the sentence definitely has too many words in it. You could just drop the word "with" from the end and it would work fine. So "with" is a preposition, and the object seems to be the earlier word "preposition", but that word is already the object of "to use" without needing a prepositional phrase. The word "with" is tacked onto the end with no structure.
I think you're correct - I knew when I wrote it that it was a little forced on my behalf. Certainly your version sounds more natural, though I have heard people say similar things to my version.
If they're talking about a specific child, sure, but if they are talking about a single, unspecified child, they is becoming more popular[1].
"Bill is over there. He is playing with his toys."
"If your child is in the kitchen, make sure they don't touch the hot stove." vs "If your child is in the kitchen, make sure he/she doesn't touch the hot stove."
For many readers the he/she breaks all flow the sentence had.
This is a false dilemma. "He/she" was never even mentioned.
In the context of the article[0], other options exist like:
If your child is in the kitchen, make sure he doesn't touch the hot stove.
If your child is in the kitchen, make sure she doesn't touch the hot stove.
These sentences are no harder to read than if "they" had been the pronoun used.
[0] - and I really think what the author is trying to do is make the reader think of a different child with each bullet point. Not a single, unspecified four year old, and not a collective group of every four year old, but one unique four year old for each bullet point. Alternating the gender of the pronoun creates this effect.
The issue is when people (seemingly) randomly switch between he/she and his/hers, even within a single context. Then you end up with something like:
"If your child is in the kitchen, make sure she doesn't touch the hot stove. If he does, however, ..."
Even when the switch isn't that abrupt, if someone is switching between he and she every few paragraphs, it can become very distracting. Using singular they solves this problem.