Why has the rule against warrantless arrests in a person's home (outside of emergencies) survived to the present day?
Where law enforcement agents possess probable cause to believe seizable property will be found in a particular location and have obtained a search warrant, the occupier of the premises has no legal right to resist the intrusion. If police have a warrant for the arrest of the home occupier, the warrant carries with it the power to enter the home to effectuate the arrest. In the absence of an arrest warrant law enforcement officials generally may not enter a private home to arrest the occupant.
The Fourth Amendment has been interpreted to prohibit warrantless arrests within one's own home, except in the case of an emergency. Why has this rule survived to the present time?
The whole point of the 4th Amendment is to prevent the government from arresting or searching people without having a good reason to do so. We do not want a government that is allowed to arrest or search people whenever a police officer feels like it. This is why the rule that you have mentioned has survived to the present day.
The basic rule for arrests is that a police officer must either have a warrant or there must be some sort of emergency that makes it important that the officer make the arrest right then, without taking the time to get a warrant. If warrantless arrests were legal outside of emergency situations, the 4th Amendment would be meaningless with regard to arrests. Warrantless arrests within a person's own home would be even more offensive because a person generally has a reasonable expectation of privacy while inside his or her own home.
So, this rule has survived to the present because A) the requirement that police have a warrant is the basis of the 4th Amendment and B) because a person's home is his/her "castle;" the place in which he or she has the greatest reason to expect privacy.