I would say this is largely up to DM discretion. This can be seen 2 ways:
Because the item summoned is replicating an item that according to the PHB has specific damage values attached to it (The longbow specifically does 1d8 as per PHB pg 149 for example), I would think that yes, the bow/crossbow is dealing damage and would disappear.
Because the attack roll is for aiming an arrow, and the damage is piercing, from the arrow, the bow itself is not doing the damage. The bow/crossbow would only be doing damage if used as an improvised weapon to physically deal damage.
The intent of the errata ruling appears to have been to prevent the conjuration ability from being used to cause damage, however, when this exact topic was addressed by Jeremy Crawford here it was stated that a conjured weapon could be used.
To my knowledge this comment was written before the next printing of the book began, which contained the correction referenced in the errata.
Taking Jeremy Crawford's comment into consideration, I would again say that this falls to DM discretion, because 2 contradictory rulings have been given from official Wizards of the Coast sources. The most recent wording of this ruling would cause me to, however, believe that the former interpretation of the rule listed with the number 1 above, is the correct interpretation of the rule as of this point.