Analogy - if it was 1950, a robber fled on foot through a residential neighborhood, and he no longer had the wad of stolen money when the police finally got grabbed him - would it be reasonable for the police to search in the shrubbery, groundhog holes, loose dirt, sandboxes, etc. of the private property in that residential neighborhood for the stolen money?

Does your answer change if, instead of stolen money, the police are looking for the robber's weapon, or the victim's empty wallet, or some other "we have good reason to suspect that he ditched it somewhere around here" evidence?

> would it be reasonable for the police to search

Yes. And my answer would not change for the other case. It's also reasonable for the property owner to demand that the police get a warrant first.

I'm not arguing about whether or not these powers are OK. I'm just arguing that, in my view, both of the situations the OP mentioned are equivalent in that they're both searches of private property.