No whether you have $1 in the bank or $100 million is irrelevant.
Your job is to understand and mitigate financial risk as is required by law.
And in this case there are many options for managing this risk other than splitting it up manually into multiple bank accounts. Speaking with a financial advisor would help with this.
Taxpayers are not covering any cost. The banks will pay for it via potentially higher FDIC insurance premiums. It’s possible they pass that on to customers, but there’s no direct link here.
Your job is to understand and mitigate financial risk as is required by law.
And in this case there are many options for managing this risk other than splitting it up manually into multiple bank accounts. Speaking with a financial advisor would help with this.