Well, I think it makes more sense when you realize police can't enter your home to arrest you unless they have a reason first, and that they must supply that reason before they can come into your house. (whether it's to a judge to get a warrant, or to the home owner to get consent to enter) If they have a warrant, they can say, "I have a warrant." That doesn't take very long. Again, usually, if they have a warrant, they'll break the door down and there won't be any asking involved.
I think the situations tend to work out a lot more poorly when you must let any police officer into your home if they want to come in (not part of our laws) allow them to arrest you first, and then explain that you were obstructing justice by not opening the door fast enough when they wanted to come in. That's kinda the chicken and egg problem for me. If the cops were there for no reason in the first place, there was no 'justice' being obstructed, hence no crime was committed at any time, except the ones being perpetrated by the police. (breaking and entering, false arrest, battery, etc)
So again to reiterate, the only time police are asking for permission to enter somebody's home is when they have no legal reason to be there. If they have the warrant, they don't ask for permission. If they don't have the warrant, that means they're fishing (trying to invent) a reason - which most likely won't hold up in court, but does effectively terrorize and harass their intended target, and also usually means they spend time in jail for doing nothing wrong. So mission accomplished, laws broken, police 'win', everybody else loses.
Again, if somebody wants to be clever, there are exceptions to the warrant requirement for entering a home - but it requires an "emergency situation", or clear open evidence that a crime has been committed in every example that I'm aware of. Police in our country are not at the "mercy" of "criminal rights." They arrest more people than any other country in the world, and I don't think it's because our country has more criminals than anywhere else.