He claims private security would not have access to the some kind of intelligence that the Met for example would have on credible threats to him and his family. But, as it has been pointed out in this forum, if the Met or the UK intelligence services had information on any credible threat against the Sussexes while they are in the UK, action would obviously be taken.
And just to be clear, as far as I understand, he is not proposing to pay off-duty Met officers to work for him. He is asking to have regular, on-duty Met protection, but reimburse the British Treasury for the associated costs.
Another interesting aspect of this discussion is that , in the US, the Sussexes' private bodyguards don't have access either for example to FBI intelligence on neo-Nazi or terrorist threats, but that doesn't seem to be an argument for Harry to demand state security. It seems that he is claiming that the threat to him and his family is somehow bigger in the UK than in California where he is currently residing to the extent of claiming in his petition to the High Court that, without Met Police protection, he and his family would be literally prevented from going to the UK.
Well, there is no way British law enforcement or the UK intelligence services (which BTW are among the best in the world) would share intelligence with unvetted private contractors. That is not going to happen.
As for foreign heads of state/government or people of equal rank, my humble understanding is that, as internationally protected persons, yes, they are assigned British state security while on UK soil, although some of them, like most notably the President of the United States, will also rely on their own national security services (in the case of the POTUS, the Secret Service) to protect them. In that case, the foreign security services would need to cooperate with the local authorities and be given permission to conduct operations on British territory where in principle they do not have jurisdiction.