The main risk with leaving an air bubble in a syringe is inaccurate dosing. With the syringe partly full of air, the amount of liquid won’t be accurate to the markings on the syringe.
Getting air into a vein isn’t a significant hazard unless it’s a huge amount – like a whole drip tube full or someone squeezes a drip bag in, including the large air bubble.
The air will circulate in the veins and reach the lungs where it will lodge and get removed in a few minutes. The lungs will filter out the air so it can’t travel to the brain except where there is also a “hole in the heart”.