As to the relationship between savings and investment, and the role of banks in there, ask yourself the following question:
Suppose that Alice decides that, from this month on, she will start putting aside 10% of her monthly income into a long-term savings account. Will this decision cause the bank to make more loans?
By default, the answer seems to be no. After all, banks lend out money when creditworthy borrowers show up at their door. How could Alice's decision possibly encourage anybody to borrow? If anything, her decision has reduced some businesses' incomes, and those businesses are now less likely to ask for a loan because they have less reason to expand their business.
If you want to argue that the answer is yes, you'll have to point out a clear transmission mechanism to justify your answer.
As to the point of why banks continue to offer savings accounts, the answer is mostly one of interest rate differentials. Banks don't have to offer savings accounts. They could refinance themselves via other means. It's just that those other means typically come with a higher interest rate.
If a bank gains extra money in the vault, won't it lower the interest rate it's offering on loans to try to attract a credit worthy borrower so the money isn't just sitting there?
Perhaps. There are two problems that significantly weaken that kind of transmission mechanism:
1. It is not clear how sensitive to interest rates business are anyway. If you have an X increase of savings, which reduces business revenue by X but also decreases the interest rate by Y, which of those effects is stronger? It seems to depend a lot on circumstances.
2. To a first approximation, there is no incentive for banks to decrease their interest rates in the first place, because they won't have extra money in the vault!
This may seem counter-intuitive, but it follows from how the banking system works: If you transfer money to bank A and leave it there, the bank is initially going to have more money in its account at the central bank (aka "reserves" aka high-powered money). This earns zero or negligible interest, and so the bank does have an incentive to get rid of it.
However, when you did your transfer, that central bank money (aka "reserves") arriving at bank A did not come out of nowhere. It came from the bank you made the transfer from, say bank B. B now likely has a short-fall of reserves (because while bank B has "lost" X amount of reserves, and the sum of customer accounts has been reduced by X, its reserve requirements have only decreased by rX, where r is the reserve ratio).
So, bank B will be happy to borrow (X - rX) reserves back from bank A, probably at current market rates, while bank A must hold on to the remaining rX due to its own increased reserve requirements. Everything is back in balance without any change of interest rates.
Things are slightly different if you put your money into a different type of account or bank bond in such a way that the sum of all reserve requirements in the banking system decreases. In that case, banks have surplus reserve requirements and will therefore bid down interest rates in the interbank market.
Under normal circumstances, the central bank - which has a fixed interest rate target - will step up and sell assets in exchange for reserves so that the surplus reserves disappear. Hence, banks still do not have any incentive to reduce the interest rate that they offer to the public!
This really only changes when the central bank decides to change its interest rate target.
What about interest rates offered to customers? It seems that those really tend to be calculated as cost-plus based on the central bank's rate target (with some long-term expectation thrown in). Plus, they probably move with some delay because that market isn't so fast. Since the target rate has been at zero for a very long time now, savings behavior really shouldn't make a difference.
Mostly though, I think it's the first point that matters: Business decisions aren't that interest rate-sensitive in the first place, at least compared to other considerations.
Suppose that Alice decides that, from this month on, she will start putting aside 10% of her monthly income into a long-term savings account. Will this decision cause the bank to make more loans?
By default, the answer seems to be no. After all, banks lend out money when creditworthy borrowers show up at their door. How could Alice's decision possibly encourage anybody to borrow? If anything, her decision has reduced some businesses' incomes, and those businesses are now less likely to ask for a loan because they have less reason to expand their business.
If you want to argue that the answer is yes, you'll have to point out a clear transmission mechanism to justify your answer.
As to the point of why banks continue to offer savings accounts, the answer is mostly one of interest rate differentials. Banks don't have to offer savings accounts. They could refinance themselves via other means. It's just that those other means typically come with a higher interest rate.