if you keep one handgun on your person at all times, you don't need to run and unlock your safe to get a gun if there is a home invasion in progress while you are home.
Unless the gun is with you at all times, there is always a chance that the home invasion might occur while you are not sitting/standing next to where you've stashed your guns. And if they find and grab your gun before you do, you are screwed.
Criminals are way more resourceful than many people give them credit. If you experience a home invasion while you are not at home and your guns are not locked up, they will find and steal every gun you've stashed around the house.
Yes, a gun safe is not an impenetrable Fort Knox and given enough time they can be broken into. But they do slow down the criminal's ability to do a grab 'n run.
A monitored burglar alarm also help to reduce your chances of being the victim of a home invasion, as long as you use it.