Why has the rule against warrantless arrests in a person's home (outside of emergencies) survived to the present day?Where law enforcement agents...

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.