In the anime and presumably the manga, we see at least one instance of Ichigo using Zangetsu to purify a soul. But how is this possible? Zangetsu is the manifestation of Ichigo's Quincy power, which is known to destroy souls rather than purify them. How does this work?
There are three reasons why Ichigo can purify Hollows.
Rukia transferred her powers into Ichigo, which would mean that he not only has the ability to see Hollows properly, but also that his general Soul Reaper powers would largely mirror hers. He wouldn't have inherited any of her abilities, but the situation would be no different than him being a Soul Reaper trainee sent on assignment in the real world, fighting and purifying Hollows.
Ichigo has these powers innately with him, given that
his father was the former 10th squadron's captain, and that would make him also half-Soul Reaper and half-Quincy.
This means that, at a miniumum, any previously awakened powers were only limited in capacity, but not directly impacted.
Quincies make use of bows and arrows (and sometimes other implements) made exclusively from their Spirit Pressure, which have the direct effect of destroying a Hollow and upsetting the balance. Only a Soul Slayer could effectively destroy Hollows.