That is an incorrect statement. UNREASONABLE searches and seizures are prohibited. That implies directly and logically that REASONABLE searches are permissible.
Warrants have never (that I know of) been a requirement for every search. Customs searches fit that situation.
Amendment IV
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
The idea that you could be searched without a warrant when it is 'reasonable' is just not true.
Obviously I have to disagree and I think that even current practice shows your statement to be incorrect.
You've got this part exactly right.
That implies directly and logically that REASONABLE searches are permissible.
And here, you are so close to having it right. You just didn't take it far enough.
Unreasonable searches cannot be done under any circumstances. Reasonable ones require a warrant AND a basis for the search. It's not a fishing expedition.
"... and no warrant shall issue but upon on probable cause, supported by oath or affirmation, ..."
There will always be those who argue the 'necessity' do to more than is allowed. But these searches are a relatively new thing. One thing or another has always caused concern for safety, but only recently -- in our lifetimes -- have we succumbed to the siren call of safety over liberty.