This is not always an either-or decision. A common arrangement keeps the Shopify channel as the primary source, because it maintains the connection and the item base with no effort, and adds a supplemental feed from a feed tool that overrides only the attributes worth changing, such as titles, product types, custom labels and a few filled gaps.
That split keeps setup light while giving campaign structure the fields it needs, and it makes rollback simple: switching off the supplemental source returns the account to the plain sync rather than leaving you with no feed at all.
The alternative is a clean handover, where the feed tool becomes the primary source and the channel app handles account linkage only. That suits stores publishing to several destinations, since one set of rules then serves Google, catalog ads and marketplaces. Either way, keep one owner for feed logic. Two systems writing the same attribute is a reliable way to spend an afternoon wondering why a title changed back.