Sure. One can avoid using gendered pronouns in generic contexts. As a writer, one doesn't have to use them at all if one doesn't want. Indeed "one" makes a fine indefinite pronoun when used properly.
My own style is to use he as super-generic (i.e. bordering on the proverbial), "one" as personal, shared person perspective generic ("one" in this case doesn't differentiate between first and third person, and has some aspects of second person imperative thrown in), and "he or she" as highly descriptive generic.
It works very well. I use "one" most often, "he or she" rarely, and "he" only occasionally. What I hate doing however is having to explain that by "man" I mean it in its original sense. That's why I prefer to say salesman over salesperson, etc. -man never carried with it gender assumptions. If it did, we wouldn't have the root in the word "woman." Now -wif did. Which is one reason why referring to an "alewife" as a female bartender is somewhat archaic.
What I hate doing however is having to explain that by "man" I mean it in its original sense
That's because the meaning has changed, and also the sexism of the term is better understood. If you're American, do you also use the term 'negro' to refer to black people? It's a neutral and descriptive term in medical parlance, but it's a clear case of a word having more impact than its roots back in the day.
So trying to shoehorn in an out-of-date definition is changing the modern meaning of the word, just the same as repurposing 'they'. In truth, I think you're just being stubborn about wanting to use sexist language, since you don't seem to have too much of a problem with using all the borrowed words and grammar we've gained since the 'alewife' days.
My own style is to use he as super-generic (i.e. bordering on the proverbial), "one" as personal, shared person perspective generic ("one" in this case doesn't differentiate between first and third person, and has some aspects of second person imperative thrown in), and "he or she" as highly descriptive generic.
It works very well. I use "one" most often, "he or she" rarely, and "he" only occasionally. What I hate doing however is having to explain that by "man" I mean it in its original sense. That's why I prefer to say salesman over salesperson, etc. -man never carried with it gender assumptions. If it did, we wouldn't have the root in the word "woman." Now -wif did. Which is one reason why referring to an "alewife" as a female bartender is somewhat archaic.