Most service weapons (handguns) the police use are not suitable to kill any large animal quickly.
As for military 306 I assume you mean 308 or 30-06 either way the military uses FMJ ammo which again is not legal to be used by hunters. FMJ does not expand and just punches a hole. Hunting ammo is designed with a soft point to allow the jacket and underlying material (lead or similar) to expand and create a more extensive wound channel.
All that said I've had deer run over 100 yards after taking a 30 cal round through both lungs and explode the heart. A bear may well use that time to disassemble the shooter if in range. Few people could chamber another round in bolt action in the time it takes a bear to cover 40 yards.
Head shots "may" be instant or in case of my recent experience with a large determined steer maybe not. He took 9 rounds of 40 S&W into the forehead before staying down and covered a couple of miles while those shots were administered. All at closer than "safe" distances. Took 2 from muzzle almost touching for him to quit getting up after he all ready had 7 holes in his head. This was out of 16" carbine not a pistol.
Normally that 40 does the job with out over penetrating like a 308...... I will have the 35 rem on hand for the next one just in case....