Particularly business loans.
For example, say I buy a home for $400k and gut renovate it for $200k. Let's say that the specific renovations are going to improve the value of the home, but it's hard to know by how much. If I go to Chase, they will gladly lend me 80% of the $400k ($320k). But Then I need an $80k down payment + $200k for renovations, or $280k cash on hand to make this work.
If I go to a smaller bank that doesn't do things as algorithmically and knows the local market, they might say "hey, we're gonna give you the $320k down payment and then we think that in that area of Boston and based on what you are doing, the house is going to be worth $150k more when you finish renovations, so we're gonna finance 80% of that $150k as well ($120k). You are still going to need to prove that you have your $80k down payment + $30k (20% of $150k) + $50k (the difference between what renovations cost and what they will add to value of home), or $160k total. So in this example, I need to have $160k in the bank if I want to make this whole transaction work with Regional Bank X, whereas with Chase, I need $280k.
If Computer Says No, nothing is happening. But Computer lacks any sense of context or local variation. By definition decisions are based on national stats which average a lot of behaviour.
Smaller banks have people - who are probably experts - making contextual loan decisions. Someone's good character and work ethic - or lack of - is going to influence the decision.
This doesn't make decisions infallible, but it's the difference between small-focus rigid decision making, and broad-focus community-dependent decision making.
It's also why so many people are caught in the rental trap. Many of them are perfectly able to afford a mortgage, but they don't match the bureaucratic criteria on some relatively minor point, and so Computer Says No.
It's also why credit scoring is so slanted. I have a perfect payment record, but no loan history because it's more than six years since I paid off all my debts. If I apply for a credit card I'll be marked down because I don't have a large existing credit limit.
This is deliberate policy, because lenders don't want people like me who will pay off the balance in full every month. They want borrowers who won't. This prioritises immediate profitability - until the loan book blows up with a wave of defaults during a recession, because these borrowers are inherently riskier.
Your anecdote presumes that the small bank is right and the big bank is wrong. I'm not so sure.
Also, whay does 'right' even mean in this case? They lost the customer. Unless customer default on the loan, they won't be 'right'
This isn't how banks work. They put in place rules that they consistently adhere to when they decide if they're going to give out a loan to someone or not. The real answer is that if consistently allowing that type of loan would make them more money than it would cost, then they won't be right.