Yes they can arrest you on private property in the manner in which you described.
There are a few issues here.
1. If you committed a crime, and the police saw you at a 7 eleven could they arrest you. yes because it is not your own dwelling, and yo have no expectation of privacy.
2. Pulling over in a private lot or driveway does not automatically require a warrant for arrest. The number one thing police do prior to pulling someone over is look for a safe place to do it. If you pull into a deserted parking lot, you can be liable for criminal
3. If the police had been in pursuit they can proceed and skirt some of the constitutional protections, such that the constitution is a shield to flagrant abuse and not a shield to enforcing the law.
If you had a bottle of vodka open in the car , turned off, in the lot, you could still be charged with having an open container
and possible DUI.
I agree with you that you should attack the merits of the actual violation, as the location of the stop based on your facts will give you no benefit.
If this helps please click ACCEPT.