Why [not have a gun in the glovebox]? I would much rather notify the officer that it is present than to keep it from him and having it under the seat sounds worse than in the glove box to me. Its easier to access and isnt just floating around in the car.
Most new cars offer more options for gun storage other than just in the glovebox or under the seat, so keeping it out of the glovebox doesn't automatically mean it's "floating around in the car."
Furthermore, in many cases, a cased gun under the seat is quite secure. I have been in a couple of accidents with a gun in the car, under the seat, and in neither situation has the gun moved from its place nor has the case been displaced significantly. I'm not necessarily advocating under the seat as an excellent storage location, just responding to your comment.
The reasons you don't want to leave a gun in the glovebox are:
1. If you are stopped, you will almost always have to access the glovebox in front of the officer. While you may want to notify the officer it is present, he'll probably be a lot more comfortable (and therefore you will probably be a lot more comfortable) if you don't have to reach toward it to get your insurance/registration/whatever. Not keeping the gun in the glovebox doesn't prevent you from notifying the officer nor does it automatically mean you must keep it from him.
2. It gives you the OPTION of not telling the officer about the firearm. In many cases, you are not legally required to inform the officer you are armed. Generally speaking, if you don't have to tell an LEO something, you're probably better off if you don't start volunteering information. Especially information that may be incriminating, may make it more likely you will be hassled,
or which may result in you sitting by the side of the road while an officer demonstrates his poor gun handling skills. Remember, while an officer may be competent to handle his own gun, it's likely that your gun won't be familiar to him. Do you want to stand nearby while a person handles an unfamiliar loaded gun? From your story, I assumed that you don't.
3. If somebody breaks into your vehicle, they will almost certainly check the glovebox even if they don't look anywhere else.
4. While a gun in the glovebox is quite accessible to the passenger, it is not particularly easy for the driver to reach it.
5. There are typically lots of THINGS in the glovebox that probably don't need to be stored in the same container with a loaded gun. Especially if the gun is not holstered or cased.
There are probably more, but that's a good start.