What if banks tend to lend out too much money because of the deposit insurance? This could have other, unintended consequences beyond the narrow scope of your comment.
It’s a nice theory, but unsupported by evidence. In the US since the FDIC was set up there have been far fewer bank failures than there were before then. In fact, having that oversight short-circuits the mechanism that makes bank runs happen, since getting your money back is no longer a function of your place in line.