flying a visual doesn't mean the radar service is terminated to start with
No, but the words "radar service terminated" mean that exact thing.
Realistically, at an uncontrolled field, ATC has cleared the airspace for me to maneuver and report my landing on the ground, I can do 3 circles around the airport for all they care, so long as when I do land I let them know when I'm clear of their airspace (on the ground). At a controlled field everything will typically fall into place on its own, when you initiage a go around, you let the tower know, they will give you further instructions... those instructions my be "fly the published missed approach procedure", they may be "fly heading XXX climb an maintain XXXX".