What most savers forget or don't realize is that deposit insurance schemes do not have the money to pay for failure of a large bank. Large banks are simply too large to fail, and if one bank fails, it can cause a chain reaction leading to all banks failing.
If the bank is very small, the deposit insurance schemes could have money to pay all account holders.
So, generally it is recommended to stick with accounts in reputable banks from which you can withdraw the money at a moment's notice.
If saving significant amounts of money, in a positive interest rate environment, it might make sense to invest into a money market fund. Such a money market fund invests into certificates of deposit and fixed-term accounts having maturity of at most a year. There is a large portfolio of various remaining maturities, in various different banks. If one bank fails, it could theoretically wipe away part of your money (perhaps 10%?), but only part, and it is likely that most major banks will be saved from failing by the government, as the deposit insurance schemes do not have enough funding.
Avoid funds that have variable-rate papers seemingly less than a year (duration), but with long maturity actually. Avoid funds that have commercial papers.
In theory, deposit insurance could give incentives to take more risk, but there will be delays in getting the money from the deposit insurance. The risk-free part of the portfolio (money market, government bonds) is not the place to take risk! Almost always, if the amount of return is not good enough, it makes sense to simply invest more into stocks and less into money market and government bonds. So, only the foolish are incentivized to take more risk in the money market + bond parts of the portfolio by the deposit insurance schemes.