Well, I've never taken a plane within the EU, but usually the border checks are
before boarding, not when you get off the plane. Similarly, the border checks are in Calais for Eurotunnel and in Paris or Brussels station for Eurostar.
You have to embark and go through border checks 30 minutes before departure with Eurostar, that's a fact. Whether you will have to answer questions is very random and doesn't seem to depend on whether you give them a passport or just French ID. My wife has a Canadian passport and a French residency permit and this does make them ask more questions though.
Eurotunnel checks were lighter when I used to take it, but that was before the Calais crisis so I would guess it has become more bothersome lately.
In any case, you just can't say it's as easy as just hoping in your car and driving to Belgium and back to fetch your beer.