Is it true that cops can use the "arrest loophole" to conduct unconsented searches without a warrant?

Say a police officer pulls a car over for speeding, and has a gut feeling that it contains illegal drugs. Cop wants to search the car but he doesn't have the probable cause needed to get a warrant, so he needs the drivers consent. Well, two things are true: 1. Anything a cop can pull you over for he can also arrest you for. Simple speeding is an arrestable offense and so he can take you to jail for it. 2. Searches of the suspect's person, as well as his vehicle if he was driving, immiediately become fair game for a police search in the event of an ARREST, without consent or a warrant being needed. So the cop asks "Mind if I search you vehicle?" and if the driver says "No I do not consent to that" the cop can just say "Then you're under arrest for speeding." Then the cop books the driver and takes him to jail, and then he can search the car legally due to the *arrest* Is it true that cops often use this "arrest loophole?"