I feel there is some confusion about linguial genders, gendered pronouns, and gender-related words here. I will try to explain, to the best of my layman-linguist ability. Bring out your salt-shaker because I am prepared to be mistaken.
First, English. I don't think English has grammatical genders in modern usage. At all. The closest, I think, is the a/an groups. More on this later.
But anyway. What is language genders? They are categories of words which decide how the words are bent. We call them genders because the standard to which all languages are compared to is Latin, and in Latin, words have actual genders. Words are either male or female. They take a word and switch it's gender. For other languages, the usage of "gender" for their word classes was just inherited, just like all other grammar terms were inherited, even when the categories are completely unrelated to genders or sex. So to make it easier to distinguish between this I am going I am going to try to remember to only call gender gender and use the terms "class" or "category" for language genders. Let me know if it gets confusing in spite of my efforts.
Anyway, so take Spanish as a first example, since this is one of the few language I have more than a passing glance into. Spanish inherited the actual-gendered word classes from Latin. They have El/-o and la/-a words, the first being masculine and the latter feminine word class. For example, Hij-o is boy, and Hij-a is girl. Rey is king and Reya is queen. The word remains the same, but a change in proposition and ending changes the word class it belongs too as well as gender it refers too.
Now look at Swedish. Historically speaking, Swedish used to have literal-gender grammar classes (maybe most Indo-European languages did?), the ending -e (or no ending vocal) was masculine and the ending -a was feminine. This still have some effect on modern Swedish in how it does gendered language of say occupations (Singer, for example, is sång-are or sång-erska, and dancer is dans-are or dans-erska) or animals (cat is katt, but female cat is (archaicly) katt-a), but here's the catch -- these, while remaining somewhat in the language, are not the actual word-categories of modern Swedish. Modern Swedish has two word classes, en-words (called "masculine", because Latin-comparison) and et-words (called "feminine", because Latin) -- and these have have nothing to do with genders. These are the categories that determine how to bend words in Swedish, both regarding definite/indefinite articles and which pronouns words get. The words for singer, dancer, and cat above are all six in the same word category, en-words -- unlike in Spanish, changing the gendering of a word between masculine and feminine no longer changes the grammatical class (grammatical "gender") of the word in the Swedish language. But the grammatical classes are still called genders, because, yeah I'm repeating myself, but yeah it's because Latin as standard.
So lastly, let's return to English. English doesn't do gender-endings, you can't change the gender of a child by adding, I dunno, an "-a" to make it a childa instead of a childe or whatever. Bachelor and Bachelorette doesn't count, because they were adopted as gendered words (from French?) and their gendering is not a result of a grammatical function of English itself. English also doesn't have more than word class. I said above that a/an is the closest they have to it above, but despite having different definite articles (which is why I think they're the closest, because of how it relates to the articles in my Swedish first tongue), the words themselves don't actually categorize differently beyond that. At most, you could also stretch it to the "the and thee" variance too, which I think is also related to whether a word begins with a vocal or a consonant, but not universally applied?
Anyway thanks for reading this rant by somebody who has no linguistical background. I'm probably wrong about a bunch of stuff, but I like languages so it was fun writing it out.