You're understandably trying to make sense of a writing failure.
Indeed, the writers messed up the Arya Stark storyline:
- The house of Black and White exposes Arya to the mysteries of the many-faced gods even though she is the opposite of being ready.
- The kindly man has the most inappropriate person for the job terminate Arya.
- Arya goes unpunished for killing Merrin Trant.
- Arya, despite her long(ish) training, and supposedly being on the run from the HoBnW, walks around in the streets of Braavos like she hasn't gotten a care in the world - out of character you could say.
- Arya is completely careless about the old woman; again, where's the training?
- Arya survives sepsis from exposing her open wounds to sewege.
- Arya heals at a super-human rate, and performs a terminator-like action sequence after having been stabbed just a few days before.
- Waif performs her own T-1000 sequence, going against probably every other rule in the manual for faceless people.
- As for the Arya-Waif fight... eh, ok, I guess I can sort of buy it, maybe.
- The kindly man tells Arya she is now no one. SRSLY? No, just no.
So they broke the plot. I can see why would want to try and fix it with a contrivance such as it being the Waif rather than Arya, but - sorry, that's just not it.