Considering we don't know exactly what went on in the mind-wiping and training sessions and what other mental conditioning things were done to him prior to the events of TWS, it may be pretty plausible that he could retain his fighting skills yet not remember every detail of all his past missions.
And again- this is my point. none of it is laid out in any kind of intelligible way. And there are several examples throughout the narrative of this type of slip up.
I don't really see how having memory wipes and then reprogramming is that big of an issue and how it could be explained without being any more convoluted.
There's a very simple way, although it involved a better handling of the story in Cap 1. I explained this in a previous post.
Three important things needed to happen:
1. At some point in Cap 1, Cap gives a wounded Bucky a blood transfusion.
2. Bucky is with Cap at the climax of Cap 1, fighting to stop the drone plane attack.
3. Like in the comics, Bucky falls into the ocean. He goes into cryo-freeze, similar to Cap, but a great distance from where Cap is eventually found.
Bucky, while spending years in suspended animation does not age- but the concentration of the super soldier serum builds in his system during cryo-sleep. Hydra, while trying to recover their equipment (And/or the Tesseract) find Bucky.
They thaw him, and find that he's developed superhuman strength, endurance et al similar to Cap. But it can't be practically synthesized, because Bucky only received transfused blood, instead of the full treatment that Cap received. Also, it isn't useful to try and send an army into years of cryo-sleep only to find the process didn't work.
What does this accomplish?
1. Bucky has no memory, without there being a need for memory wipes.
2. Bucky is a one-off super soldier. Thus a reasonable explanation for why there isn't an army of Winter Soldiers, which the film doesn't give.
3. The process ******s aging like Cap.
No convolution at all. A simple and dramatic origin for TWS.