I thank you - and all other who replied - deeply, but I'm not sure I understand you - or agree with you.
I do assume there are two kinds of banks. One is "retail", or financial services, as I understand it. Gets its money as fees. Another is investment - this one takes risks and reaps rewards, but it's a different kind of organization - somewhat similar to a group of people, who pooled their money and work on a kind of gambling, often reaping rewards, but also taking risks.
Talking about this second group, it's entirely different - for many people who are not members of that group of investors (risk-takers) it's outside of what they deal with. So I'm mostly talking about first group.
For example, my point regarding insurance is that I'm dealing with professionals having different amount of information than I do, and benefiting from that. Theoretically market should bring insurance premiums to some average profit margin. Also theoretically I should be able to earn on average more than I spend on average on insurance, so examples with big expenses are supposed to be exceptions, and on average I'd have more money paying for cases myself. In that sense, insurance premiums are spent (supposed to be, mostly) on cases where insurance events happen.
Next, "those same deposits" - I think the word "principal" is meaningful :) as there won't be interest without principal. So I still think we talk about deposits - together with added relatively modern service of investing, getting some averaged interest and paying some fees to the service which does the investing. Yes, that's the core service. Of retail banks... may be we actually agreeing here?
Next, of course banks invest, that's how they pay that interest. The principal money still aren't theirs - so banks don't "fund" - as in "fund with their own money" - since retail banks by definition don't deal with their own money; instead banks pass those money as credits - or, yes, even to buy shares. The fact that, say, universities takes their budgets from interest shows that service works - but it's not the banks, who "pay" - don't assign the source to them - it's principal, combined with the service (for which banks don't pay, but receive fees), which generates interest. Bank is a sort of an engine which you feed with fuel to get desired outcome (bad analogy, I know), but I think fuel here is more principal (sic), more fundamental than the service.