I work at an academic hospital with a cardiothoracic surgery training program.

Depending on the case, the attending does not always need to be present the entire time. A routine aortic valve replacement, for example, has multiple steps, some of which are more critical than others. Once anesthesia has the patient asleep, a senior CTS resident or fellow should be perfectly capable of cutting through the sternum with a bone saw and incising through the pericardium to expose the heart and great vessels. Once the attending surgeon arrives, the resident will place+connect the cannulas for cardiopulmonary bypass, and the perfusionist will deliver cardioplegia (to arrest the heart) and begin the pump run. The surgeon then joins the case hands-on for the team to perform the critical incisions on the heart and aorta. For more complex operations, sometimes the attending will call for another attending to join the case (for example one surgeon may be comfortable with the aortic valve, but may call a colleague to help if the patient needs a mitral repair or a septal myectomy). Once the aortic valve prosthesis is in place and the heart is stitched up, the team will return the blood from the cardiopulmonary bypass circuit reservoir, restart the heart, and ensure the heart has restarted in stable condition - the anesthesiologist will look with a transesophageal echo probe to see if there are any problems. If everything looks okay, the attending will leave and the resident (maybe a different resident if the first one had to leave) will wrap up, close the pericardium, close the chest, and suture up the incisions.

So the case is done in continuity, but the attending surgeon only really needs to be there for the critical part of the case. What’s “critical” vs. what’s not depends on the case, of course, and if there is significant troubleshooting because something goes wrong then that obviously requires a longer time.

TL;DR people rotate in and out of the operation to execute their particular roles when they are indicated. Not sure how it works in other specialties but I bet it’s something similar.