There's plenty of room for improvement on the FSS. The "snapshot when entering" idea is one, but i would prefer something completely different:
1. "Snap to target"
When your marker is close enough to a target (range can be adjusted in settings) and the speed you apply to the marker is low enough (speed limit can be adjusted in settings), the FSS automatically locks the marker to the object and follows it.
As positive sidenote, this would not only help when trying to catch moving objects, but also would make the FSS generally a bit more enjoyable to use.
2. "Toggle through contacts"
When your have several resolved (means, full circle) objects in your screen, you should have a "toggle" button available. If the marker is on no object at the moment, it jumps to the nearest not scanned object. If the marker is on an object, it jumps to the next not scanned object.
Bonus: a second toggle option could also go through resolved objects. But in that case, things would be much more complicated to implement. The game would at the moment of using the toggle the first time create an internal list of all objects in range, then go through them by button presses.
3. "Overview"
Please finally give me my ships radar somewhere in the FSS menu. I want to have a chance to see what's happening while i am in the scanner.
And of course, as suggested in the OP, eliminate the "throttle at zero" requirement.