PDR1
Your explanation sounds convincing, but is it actually true?
The air initially has zero momentum in the fore-aft direction.
The fan draws in air thereby giving it some fore-aft momentum.
It the nozzles air pointing directly downwards, the air leaving the engine has zero fore-aft momentum.
The overall change in fore-aft momentum is zero, so there will be no resultant fore-aft thrust force.
Or we could say that the fan exerts a rearwards force on the air giving it a rearward aceleration. But the rear engine casing between the rear nozzles and the nozzles themselves, exert an equal and opposite forward force, reducing the rearward air velocity to zero.
The overall fore-aft forces and acelerations sum to zero, so once again we have no fore-aft resultant force.