For exactly this reason, US financial regulators recognize Bank Holding Company (BHC) as a separate category from chartered banks. BHCs are subject to a more limited form of the examination banks receive, in order to ensure that they are not in danger of going insolvent (leading to the insolvency of their bank subsidiary). But the examination for BHCs is nonetheless much lighterweight.
Interestingly, all BHCs are regulated by the Federal Reserve (which is not true of banks proper, which may "choose" their regulator within certain limits from the FRS, OCC, and FDIC, occasionally others). There are some advantages that come from the FRS's approach to BHCs, including better access to FRS lending. As a result, many "banks" today, in terms of consumer branding, are actually BHCs that do all of their actual banking (depository financial institution/DFI) within a subsidiary.
"Bancorp" is a somewhat informal term for BHCs, so "banks" with bancorp in the name are often (but not always, the term has older uses) actually BHCs.