It is not an exception to the Constitution, it's a decision about what “unreasonable” in the Fourth Amendment means.

Note that the default (but not universal) equirement to get a warrant for a search or seizure (and the imputation that for many warrantless instances of either, probable cause is still required) is also such an interpretation; the text of the Amendment doesn’t say either of those things, but they have been inferred by the Supreme Court to be generally the case from the juxtaposition of the reasonableness requirement for searches and seizures and the probable cause requirement for warrants.

While the Bill of Rights (and protections in later amendments) is sometimes treated like a bit of divine revelation, much of it is intentionally (to kick the can down the road on resolving disputes at the time) imprecisely worded, heavily compromised, legislative enactment by imperfect legislators, with sentences that are disjoint and where any meaningful application requires reading connections into the the text that aren’t explicit, as well as devising concrete operationalizations for vague terms like “unreasonable” or “due process”.