> 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.
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?