Sample 3rd person - He She or They?

I find a few american non-fiction authors use 'she' when referring to an example 3rd person of indeterminate sex.

e.g. if a person exercises then she will stay healthy

instead of the more typical

if a person exercises then he will stay healthy

It seems to be a deliberate attempt to be 'fair' and demonstrate their 'awareness of gender issues'. The problem is that when the topic is complicated it breaks the flow of thought and causes you to stumble and think about their promotion of 'gender equality' which is kind of a pain.

It seems to be American authors who mostly do this.

In informal speech it is common to use they

e.g. if a person exercises then they will stay healthy

I don't know if this is grammatically incorrect but at least it is gender neutral and would not cause the reader to pause and have to digest the author's (dare I say tedious...!) attempt at political correctness.

So questions:-

1. Is 'they' grammatically incorrect?

2. Can 'he' be considered more grammatically correct then 'she' due to its lack of intrusion into the flow of the dialogue and traditional usage?

Not trying to be sexist ladies(!) Just would like to focus on a topic without sanctimonious PC distractions being peddled every few paragraphs.

If 'they' is incorrect could it not be a candidate solution to a grammatical problem and so be a candidate for being legitimised?