TBF, I agree that they've had some horrible comments addressed to them and that these were fuelled by racism. And if the police identify a threat, of course they should pass the info on (for free!).
However, my first thought was similar to AllThePogs in regards to DV victims and that common people really wouldn't have access to what he's asking for even if they're at significant risk. (I know a different poster said she had more intervention though).
When my ex's behaviour was seriously escalating the police agreed that my son and I are at risk, with my son being at risk of kidnap as well. And then basically just expected me to get on with it and curtail my life and... that's about it. Nothing else was ever offered- this is basically the norm as well.
And there's much, much worse cases than mine out there and there's never a thought that this would be the appropriate level of support, especially as most people can't afford to pay for a private company either.
And no, the private company shouldn't be allowed guns either. Harry may well choose reasonable, well-trained individuals who would only intervene if absolutely needed. There's no guarantee whatsoever that every other individual rich enough to afford security detail won't hire some trigger happy goon who would shoot if you so much as look at them.