Has grammar truly gone out the window?

When the plural is used but no specific numeric indicator is given that would explicitly include “one” then “one” is just as valid as “more than one.” Singular is used if and onlyif you’re talking about one andonly one of something.
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The above was used to try and refute evidence presented in another argument, but I don't think that arguing grammar really had any meaning in that discussion. Now, while I understand that sometimes we are required to use plural nouns in place of singular nouns to make something grammatically correct, but isn't that when we are speaking about something where there are a variety of different ways something can be categorized or something? I'm not really conveying the thought I'm having well on the issue, but I'll try some examples related to the sign above.

"No dogs of any kind are allowed." Is this not because there are many kinds of dogs, and that because all dogs are not allowed, we must use the plural? No dog allowed is simply incorrect when it comes to grammar, and thus cannot be written.(I've just become paranoid that I will make a simple grammatical mistake somewhere in this post now and for that I will chastised. I suspect it to most likely be, if anything, a dangling participle or ending a sentence with a preposition.)

I am aware that when we speak of all of something, that we use plural nouns. And while in person, and writing, I tend to use "they" to refer to singular individuals, I use it because it is a single word that conveys I don't know the gender of that individual being discussed. If someone has a word that would be better suited, I'm all ears, and though most people seem to use this same custom, I have been chastised for using it as well.

Getting back to the point, though, is it valid for someone to declare that in all instances, a plural that is not preceded by a numeric indicator can mean one or more? I do not believe this to be the case, in so far as a plural allowing for a singular possibility is the exception. As above, the sign is saying "All dogs are not allowed," which would mean that even one is not. However, if you are looking for the parents of a child, while you may only need to find one, you are, nonetheless, looking for multiple parents (specifically two). Is this not accurate? I'm having a hard time trying to rationalize the idea that talks about requirements for the parents of a child meaning that only one need meet said requirement, despite it being pluralized.

If a school were to have admission requirements stating "Any child admitted to this institution must have parents that are employed." Would that not be indicating that both parents be employed? If it were meant to be one or more, would it not be something more akin to the following sentence? "Any child admitted to this institution must have at least one parent that is employed." Other things that come to mind are "having a parent that is employed" and "one or more parents."

Naturally, now all the quotations are making me concerned about my grammar being critiqued. I suppose I will just have to live with that, should it come up, so long as anyone that speaks ill of my grammar also comments on my inquires.