Noting first that it would be correct to say both Cincinnati and Seattle have Children's Hospitals, how sure is anyone that however widely the form is used, Children's Hospital is in fact the name? I ask because "everyone knows" the most famous children's hospital in the world is Great Ormond Street Children's Hospital… only it's not. It's The Great Ormond Street Hospital for Sick Children
Without having too much idea what "a company for children" means I suggest it is not the same as "an old people's home" unless it is literally a children's home.
Acknowledging that there should be no grammatical difference between a "children's home" or the more widely-encountered "children's charity" or, yes, a "children's company" I seriously suggest that idiom overrides that.
Going back a way, a tobacconist's shop is often called a "supplier of smokers' requisites" but when was the last time anybody called one "a smoker's shop"?
The normal usage would be none of the above but rather to name whatever is actually being produced or sold: a toy shop; a school uniform factory; a stationery supplier; a cycle shop; a sweet shop…