>> If for example I happen to stand in front of my window and witness a crime being committed in the street,
That is probably not an instance, where a search warrant would be reasonable. But if for example you denounce someone for embezzlement, and during the investigation it comes to light that you have profited from this act of embezzlement yourself, a search warrant would be given (against you) despite of you beeing a witness.
>> they could search my house without any consent or warrant?
They do need a warrant, handed out by a judge. The idea is, that a judge gets a request for a warrant and decides if it is reasonable or not. The problem is, that the judiciary is so overwrought, that almost any judge will decide in favor of a warrant, even if its not reasonable.
I am almost certain, that this warrant would not be reasonable if a judge with time had thought about what actually happens here.