It's prohibitively expensive and logistically challenging to build on a mountain, and mountains on Earth don't go nearly as high as planes can.
Florida's air is humid and dense. You get lots of heat gain from that. The density itself matters. Condensation is a huge source of heat. Freezing is another huge source of heat. So you boil off cryogenics and maybe worse. We lost a space shuttle due to a chunk of ice hitting the wing. Falcon 9 rockets need unusually cold filler; if it warms then it expands and won't even fit in the rocket. It probably pours out and gets drained away, the alternative being a rupture.
The USA would do better launching from near Tuscon, Arizona. It's decently high up and dry, which helps keep the cryogenic content cold. We used to do this in fact, before the Mexican embassy complained about defective rockets crashing in Mexico. We chose to move to Florida instead of making a liability payment deal with Mexico.
Not a dumb question at all. It would actually give you some non-trivial advantage - but altitude matters considerably less than speed when trying to achieve orbit.
Starting a mile higher would allow you to avoid the densest parts of atmosphere, giving you less air resistance and helping with the Max Q, or maximum dynamic pressure (rockets sometimes need to reduce their engine output until they have cleared the denser atmospheric layers, or they would break up due to air resistance stresses).
This does unfortunately not outweigh the cost and complication of moving your launch site on top of a mountain. If you're really after every scrap of advantage, moving your launch site closer to the equator is way better than moving it up (until you already have it at the equator, that is).
http://www.zero2infinity.space/bloostar/
The low pressure lets them get away with toroidal tanks, pressure fed booster engines, and a relatively large faring. Or at least that's the plan, they haven't launched anything close to orbital class yet.
Also, the best places to launch from are near the equator where there is a large body of water Eastward. And there are very few mountains in such spots.
But what is the point of a space elevator then?
Every time you get some altitude, you also get a minuscule amount of velocity because the Earth is rotating. A space elevator gets you so much mind blowing altitude that you'll also get all the velocity you need.