I'm far from an expert on this but my understanding is similar. I think there are several technical and market reasons for this. First is the simple cost of developing and certifying a new general aviation engine. That cost must be covered by sales in an already cost difficult market. The second part is technical...plane engines are used very differently than car engines. They are not purely steady state but they are typically used in a near steady state way. This makes improvements such as fuel injection less additive beneficial. Ad in that most of the uses of the engines are non-commercial and non-life critical and you have a logic - why would I spend tones more to gain a few points of efficiency on something I use for fun that already costs me an arm and a leg that works. Its the same reason you don't see the most fuel efficient innovations start in sports cars.

example: look at the Lycoming O-360 and the list of planes it is used in.

https://en.wikipedia.org/wiki/Lycoming_O-360

Typically they are equipped with either a Continental O-300 series engine from the 40s, or a more modern Lycoming O-320/360 from the 50s. They hark back to an era of magnetos and manually adjusting the fuel mixture. On the plus side they're so mechanically and electrically simple that there's very little to break. On the downside they're pretty inefficient and carburetor icing is still an issue. Also, they need leaded gas still.

https://en.wikipedia.org/wiki/Continental_O-300

https://en.wikipedia.org/wiki/Lycoming_O-320

https://en.wikipedia.org/wiki/Lycoming_O-360

I'm not kidding about the efficiency either. The Lycosaur O-360 is a 5.9L engine that produces at best 225hp. You can get 245hp out of a 2L Ecoboost in an economy car today. This isn't an apples-to-apples comparison, but it gives you a sense of how far engines have come since the 50s.

A Cessna 172 retrofitted with a modern engine could probably get more than a 50% increase in range, and also cabin heat.

>I heard that airliners can't land with a full load of fuel and they have to dump it to land in an emergency

I think the main reason they do that is that way there's less stuff to burn if there's a fire or explosion.

Not all airliners can dump fuel. Some have to burn it off to get under maximum landing weight, or just risk land over the prescribed MLW.

Jettisonable batteries (or lift magazines of battery + motor) are one design that's been explored. Take-off and initial climb is a major fuel-demand flight phase.