Outsourced bookkeeping services absolutely can integrate with accounting software. We have clients who, most of our clients are QuickBooks, some of which are desktop, some are online, but we also have clients who are using other accounting software. Whatever the client needs are, the outsourced bookkeeping service should be able to work with them.
If they can’t, then it’s not the right fit. They need to find, the client needs to find somebody else who can do that work for them.