Earlier quoted context omitted.
please cite a source for "The male gender is used for the gender neutral case". as far as i can tell, this is an issue with no clean solution in modern English. 'he or she' is the most reasonable option in my opinion. it is cumbersome, but that is better than sexist or grammatically incorrect. 'we are looking for a technical lead. he or she will have three years experience as a lead developer or similar role.'
Singular third person "they" is widely attested and there is no grammatical problem with it at all. It's even in Shakespeare. You can use it, and if anyone complains about it, don't hire them.
Also, citation needed for using 'they' as singular. Apart from the fact that it very strongly sounds plural, one of my personal pet peeves with English is the fact that we do not have a singular plural differentiation in the second person. As long as we are re-working pronouns, lets not repeat that mistake.
When I am trying to be careful and gender neutral (for something that needs to be speakable, so no (s)he), I try to structure it to use either 'one' or 'that person' in place of (s)he.