When the engine is cold; take out the spark plugs and inspect their filaments. If a spark plug's filament is covered with black goo, is very dirty and smells like petrol then there you go; leakage in the cylinder which means worn pistons and valve guide.
Spark plugs are like the heart of an engine and you can tell a lot about the condition of an engine by simply inspecting them!
If all plugs look good and all cylinders have good compression then the problem is not cylinder or EFI related assuming your car has fresh air / oil filters and you use good quality fuel from a reputed petrol station.
And if that's really the case; then it MUST be your catalytic converter:
Here are five signs that something may be wrong with your catalytic converter:
- Your vehicle's fuel efficiency suddenly drops.
- Your vehicle doesn't accelerate when you step on the gas pedal.
- Your vehicle may refuse to start.
- The MIL or Check Engine light comes on.
I'd rule out the transmission and axle because you're not facing any problems while changing gears or witnessing weird rattles in the chassis and this tend to only happen with cars that were in an accident or have extreme mileage.
Lastly; its very hard to give a definitive answer without inspecting and driving the car personally and I'm no engine expert in the first place (Just an enthusiast) so take everything I said with a grain of salt!
In any case; my bet is on the catalytic converter because its prone to failure after a decade of usage and your symptoms also point at its direction! But that's just a guess.
Good Day!