Support for EU within EU is growing since the war in Ukraine and has gone to overdrive since Trump 2.0. No current political party except for fringe parties in any EU state advocates for exiting the EU or ending the four freedoms. It’s reasonable to say that yes, EU citizens do approve of freedom of movement in EU. They probably do want to limit freedom of non-EU citizens though…

… which is exactly why the EU would terminate agreements with Switzerland if we start first. And why it would make political sense. They made that quite clear with the UK.