There being alot of them is, I think, precisely how they became so wealthy. Until the modern financial era, banking was predominantly built around personal relationships, and especially family relationships.

A bank in London had no reason to cash a check drawn on a bank in Paris, or value a Letter of Credit given by a bank in Frankfurt, unless the London bank trusted the other bank would honor the liability, and honor it timely. This was especially true the larger and more complex the transaction.

As a banking consumer you wanted to use a bank that would honor instruments from and have its instruments honored by the most and the most important banks across Europe. Those banks were often part of a consortium of family-owned banks, because under economic stress the family-owned banks would always honor their commitments to each other, and offer each other low-interest loans to help another member bank stay afloat.