Well IF the bar was left disconnected from the A/C and the tug that's a groundcrew error. However, the fact that the flightcrew could NOT have seen tug, TOWBAR and pin is the error on their part. Note error - not blame.
Yes 25 minute turns are challenging - that's why you have to be meticulous. So what if someone pushes ahead of you or get's taxi clearance ahead of you - big deal, it'll cost you a few minutes - chill out.
When the ground crew is released it is imperitive that they are observed
crossing the line CLEAR OF THE AIRCRAFT with the
tug, towbar, headset man and pin. ONLY THEN should the after starts be done.
BA had a tug breakdown as it pulled away. Flightcrew didn't notice because they had "released" them (night time) - a/c taxied forward and dragged the No2 pod over the lifting booms of the tug...... probably with groundcrew scattering!
Don't rush, look, be careful, check EVERYTHING clear.