EMB-175 Check Airman reported a critical ground conflict during takeoff initial climb with a taxing aircraft crossing departure runway.

Narrative

Departing ZZZ this morning Tower issued our takeoff clearance [Runway] XXR/X2 as an Airbus in front of us was on its takeoff roll. I taxied the Aircraft into position on the runway centerline held the brakes and waited until the [departing] jet had cleared the runway; before starting our takeoff. Normal takeoff roll through 80 kts.; v1; rotate for us. Climbing through 600 ft. I looked down our aircraft nose and to the right of us and saw [an aircraft] crossing the runway below. I immediately asked Tower if an Aircraft was crossing the runway in front of us? He replied with 'yes; sorry about that; contact departure; have a great day'. We continued to ZZZ1 uneventfully. I believe [critical ground conflict] was due to the fact the Controller had Ground and Tower frequencies combined and completely saturated with taxiing; departing; landing; and repositioning Aircraft.That much traffic in the morning demands multiple controllers on multiple frequencies monitoring and directing traffic safely.