I find it difficult believing that pure friendship exists between a man and a woman

Guess I'm weird then.
Not sure what sort of in-depth analysis you want. I tend to find it easier to platonically bond with women.

Yeahhhh because gay men and women don't exist..... /s
Seriously what the ****
Someone needs to stop watching When Harry Met Sally. God, these people!!

If a man has more female than male friends and hangs out with them every week, it is weird.
If a woman has more male than female friends and hangs out with them every week, it is also weird.
If you disagree, I hope you would not mind providing an in-depth analysis.

How to satisfy that they are merely platonic?

How to satisfy that they are merely platonic?

I don't think it's a clean-cut answer... IMHO it depends on the type of person and environments they've been subjected to.
For example, if you go to Uni and are in a mixed sex hall or residence, and doing a course with a reasonable 50/50 male / female ratio (e.g most social sciences), I don't see why you wouldn't get on naturally with people of either sex. Furthermore, in a Uni environment, you're constantly meeting new people... so for these people it's natural for them to appreciate the personality behind the gender. On the other hand, if you left school / education at an early age, and worked in a typically male or female dominated environment (e.g. a car mechanic or hairdressers), then of course it's going to be harder or less natural to form a platonic friendship with the opposite sex.
Besides, there are certain personality types who may naturally get on better with people of the opposite gender to them (e.g. tomboys, mummy's boy etc.)
The bottom line is you can't.
IMHO, a purely platonic friendship means that there is absolutely no physical attraction on either side.
I will concur that there are many close male / female friendships, where one person secretly likes the other. Granted, they may accept that it's never going to happen (for whatever reason)... but, as you say, if there's an element of attraction on either side, then it's not a platonic friendship, is it. Although that's not to say that a valuable friendship can't exist between the two of them.
Still, it always makes me laugh when people (particularly girls) say, "of course guys and girls can be just friends"... only to hear about how they ended up sleeping with one of their guy-mates a couple of weeks later

Maybe my friendships with my male mates only work because I don't want to shag them, either

Guess I'm weird then.
Not sure what sort of in-depth analysis you want. I tend to find it easier to platonically bond with women.

Are you a man or a woman?

a tathagata has no gender