It's not that the logs themselves increase the integrity of banking transactions, it's that they find bugs that increase the integrity of banking transactions.
Say that you've built your system like any good highly-available system so that you rigorously check consistency of the program invariants, log if there are any anomalies, and then either try to recover or back out with a user error if something goes wrong. You've configured your monitoring to alert if any of the consistency checks fail. Now you get an alert that you've errored out and served a 500 on one in every million requests.
For Reddit, Facebook, Google, or other free services, it's no big deal: one in a million means that you're serving a couple hundred, maybe a few thousand 500s per day, max, and then the user just refreshes and gets over it.
But in a financial transaction app, this is a big deal. The basic assumption you have to make is that if you have a consistency problem that your monitoring caught, you probably have consistency problems that you didn't catch. And the right thing to do is investigate until you understand what is going on and what the impact is. So you pull the logs of all surrounding requests, you pull the logs of any requests that were in flight at the same time, you pull the logs for the user that initiated the request and see exactly what they were doing. And then you try to reproduce the problem, test for it, and fix it.
The difference is entirely in the tolerances built into the system, and how that dictates that you respond to errors. In some domains, a one-in-a-million error is a "well, we'll catch it next time" event. In others, a one-in-a-million error is a "drop everything you're doing, fix it, and ensure that no similar errors exist" event.
BTW, the strictest tolerances usually are not in B2C companies, they are in suppliers of large B2C companies (or the government). The bank is completely willing to risk a few hundred dollars per customer on an ATM card, but if the ATM vendor loses that money without it being part of the spec they gave the bank, they've lost the contract. Most of these enterprise B2B contracts are based on trust, and if the customer observes you losing money accidentally, they'll wonder what else you're doing accidentally.