No one here has shown that the police in Watertown did anything but that.
Actually, in several threads you have been provided with case law controlling fresh / hot / immediate pursuit, as well as exigency. You apparently decline to avail yourself of their instruction.
The law of the land is the US Constitution, which guarantees that We are to be secure in our persons and property. The exceptions carved out by case law are narrow and specific.
The searches in Boston were anything but narrow and specific.
You argue in a frustrating way. You dodge questions, misdirect topics, answer in generalities and make vague statements like "according to the law. It's simple".
To me, this is indicative of someone who simply wants to believe a certain way, arguments to the contrary be damned.
I will just caution you that, in my opinion, you cannot see the forest for the trees here. I could have used your position to justify thousands of searches during my career; searches that I would have considered unconstitutional and morally reprehensible. As a law enforcement officer, I felt it was my duty to protect the people and their rights. You seem to omit the second part.
The right to live is the supreme right. The protection of that right when it is threatened always overshadows the demands of the lawyerly. It’s natural.