I wouldn't personally drive it until the coolant circuit is bled properly.
Try the following;
Park the car nose down so the expansion tank is at the highest point.
Drain all the coolant and start again. Turn the heater dial to 'Hot' (very important as it opens up the heater matrix and allows coolant to circulate).
Open all three bleed nipples and fill the circuit SLOWLY, closing each nipple once you have coolant running from each.
Squeeze the engine bay hoses (still with the cap off) to force coolant around the system-I find doing this with the cap off helps force air out the expansion tank.
Run up the engine with the heater on position four until the blowers feel hot. Leave the cap off until the coolant level starts rising. Squeezing the hoses again while the engine is running up will also help.
Keep an eye on the temperature gauge and feel all hoses including the radiator hoses to make sure they're getting hot (rad hoses may take a while until the thermostat opens). I find it helps to bleed quicker by raising the engine revs to around 2,500rpm (gets the water pump spinning faster).
Once the level starts rising and the hoses are hot, refit the cap, monitor the level and temp gauge and let the engine run until the rad fan cuts in.
Turn the engine off once the fan has switched off, let it cool and recheck the level.
I've used this technique loads and never failed
