I think you may have it backwards. CDs and money market funds are actually liabilities for the financial institution. They're ways of paying customers more in the hopes of attracting deposits.
There are non-depository financial institutions, but you'd be more familiar with them by other names: investment banks, insurance companies, pension funds, mutual funds
Yes, they're liabilities just like deposits. But if you offer them and not regular deposits (like the bank I mentioned) you don't need to built out a whole infrastructure for payments, checks, etc, just for the much more limited set of actions you need to support those.
There are non-depository financial institutions, but you'd be more familiar with them by other names: investment banks, insurance companies, pension funds, mutual funds