Inert is one of those relative terms that can give us lots of confusion. Some materials are slow to react with other stuff and we can call them inert but under other conditions they will react.
But what I see in the picture looks like a mix of various materials rather than a pure beach sand. So some of it will react more quickly that other parts. Unless you have extreme PH, I would not worry a minute about using it. As for the worries about it being nasty, there seems to be two ways to look at that. Being around people it may be nasty, that is true. But then if it has been chlorinated heavily, that will have sterilized it very well and the chlorine will have gassed off. I see no more reason to worry about this sand than something that has been picked up, graded, loaded and processed on all kinds of machines that leak oil.
It seems you have done the due diligence to say you have tried! Go for it and enjoy.