(help a writer)Could a large investment bank be privately owned?

I write pulp fiction and am wanting to know if this parallel universe is plausible or even possible. I am economically illiterate.

In the 90s investment banking was more fragmented and several British banks still had major family ownership. Firms floated to enable partners to access capital, is it possible that one of those families (in a parallel universe) bought out the partnership rather than listing?

I understand there are next questions of competing in a consolidating environment. Could you simply incentivise the partners with cash instead of shares? What is the difference between a float where partners sell down all of their equity and a transaction where the partnership is bought out?

Say the Kleinwort family bought (assuming they had the money) out the partnership instead of selling to Dresdner could the bank function as a private company?