would it save a worthwhile amount of energy/fuel in, e.g., a Falcon 9 launch?
As it stands? no. You'd need a whole new first stage capable of surviving the heat and pressure of launch (hands up now, who likes the idea of flame-grilled tanks of liquid and possibly rather cold rocket fuel?) as well as engines that can operate efficiently in a very hot and high pressure environment.
More importantly, you'd need to build a whole new set of launch facilities, cobble together some prototypes, blow a bunch of them up, get some working, then sell launch capacity at a discount to encourage people to use your unproven and risky new launcher, etc.
In that regard, whether or not it saves a few percent of fuel is rather counterbalanced by the enormous up-front cost and risk. It isn't a great trade-off.
Also, one could easily devise other ways of jump-starting a rocket in a rocket fuel-independent manner (e.g., dump flywheel energy into a railgun-like propulsion mechanism) that further accelerate the rocket. Why aren't people doing this? Are such endeavours simply not efficient enough to pursue?
There are already air-launched rockets of various colours and flavours... the first one that comes to mind that's used to put actual useful payloads into orbit is the Pegasus. There are more famous suborbital things like SpaceShipOne, of course, and also the likes of the ASM-135 anti-satellite missile, which was never used.
Ground-based launch assist systems seem to be perpetually academic exercises, but maybe one day we'll get some kind of maglev first stage. The MagLifter idea popped up in 1994. Non-maglev test tracks also exist.
Thing is, boring old-school staged liquid fuelled rockets work. They're reasonably priced, and reasonably reliable. There's not enough incentive to play with alternatives, yet.
For something that is a little bit like what you're thinking of though, you can look for a popular but fortunately little used style of rocket: the submarine launched ballistic missile.
To quote from the Titan II launch process described here:
TRIDENT missiles are launched from the submarine by a steam generator system. A small, fixed solid-grain gas generator is ignited and its exhaust directed through cooling water into the base of the launch tube. The missile is ejected from the tube, through the water, and to the surface. At that point, the missile's first-stage rocket motor ignites and sends the missile on its way
Note that the gas generator isn't part of the rocket itself; that saves a bit of mass and additional staging complexity.