The problem is that we are fighting the laws of thermodynamics. Any jet engine is based on the rapid expansion of a gas. Any such expansion is based on generating heat.
We do know many chemical reactions that create heat. But none of them can do so as efficiently as high density carbon based fuels. And those that do come close still emit noxious molecules into the atmosphere.
At it's core, high density carbon - fossil - fuels have an incredibly high potential energy delta. A gallon of jet fuel packs a huge amount of punch. Per the law of conservation of energy: that punch must come from somewhere. It's literally sunlight which was converted into carbon form through biochemistry. Tectonic activity compressed those carbon molecules into an extremely dense form.
Incredible amounts of energy have gone into that natural process which spans eons, and no synthetic process can hope to recreate that in an economically viable manner. The latter meaning: production here and now, in enough volume at an affordable price.
So, any alternative is competing with an energy store or sink (it's not really a source since it doesn't create anything!) which is extremely portable and yields a high potential.
Moreover, since the invention of flight, only 2 main modes of active propulsion have emerged: a gas-based exhaust (jet engine) or propeller based.
Going electric means ditching the jet engine. But then you end up with a less efficient propeller based plane. The main reason why commercial airlines ditched propeller based planes for jet engines is because the latter can efficiently keep a large load (people / cargo) in the air.
Batteries don't pack the amount of energy needed to go long-distance, plus their weight makes battery-based flight not really a viable option. Plus, that's just shifting the problem: the energy still needs to come from somewhere.
Solar array based flight has potential, but solar cells aren't anywhere near efficient enough yet. Current solutions combine solar-based propellers with gliding, which result in large span widths. Whereas they are nowhere close to being able to lift the load of a 767.
The U.S. even invested and developed nuclear jet engines. No joke! Look up Project Pluto: a nuclear based ram jet engine. The idea being that nuclear power would rapidly heat air. They even build a bomber (NB-36H) that carried a nuclear engine in hopes of one day hooking that up to a jet engine. Kennedy cancelled that program because he believed that it was all just too provocative in a Cold War era.
So, the strategy on the table is finding ways to make the combustion process itself (a) more efficient (b) less wasteful. It's a delicate balance and it's by definition a limited balance. Inevitably, the gains will get smaller and smaller until it's not economically viable anymore to keep researching them.
To my mind, I feel we're already past the largest gains that we could make in terms of fine-tuning the technology. And to add insult to injury, the gains that still can be made are always offset by the increasing demand of more/cheaper flights / destinations.
The hard reality is that flying should be far more expensive to even remotely approximate the impact on the atmosphere. The other option is that everyone should be able take a plane like they take a bus (democratization), and accepting the downsides, even when those contribute to our own demise.