You only need and should keep enough liquid cash needed to handle day to day withdrawals. Everything else is fair game up invest.
See: https://en.m.wikipedia.org/wiki/Fractional-reserve_banking
“[...] each tetherUSD in circulation represents one US dollar held in our reserves (i.e. a onetoone ratio) which means the system is fully reserved when the sum of all tethers in existence (at any point in time) is exactly equal to the balance of USD held in our reserve.”
That said they don't state anything about what their reserve is. Wealthfront has a 2.3% savings account for us normal people. If I created a stable coin like tether is and had a billion dollars of investment; I could make 23 million dollars a year and tell people that I have a 1:1 backing without lying.
If I went to a bank with a billion dollars I could likely negotiate an even higher rate for such a savings account.
I don't think that's how it works. As others have mentioned, FDIC insurance only covers 250k per depositor per bank. Typically when entities have that amount of cash (think Apple, Google, etc.), they put them into US government bonds, which are effectively equivalent to deposit accounts.
Edit: this is an interesting point, though. Ignoring the insurance issue, what would a bank do if you asked to open a deposit account with a billion dollars (paying, say 2%/year)? My uninformed layman's speculation would be that they would reject you because they wouldn't know what to do with that much money: they have to somehow lend out enough of it to at least break even on the 20 million/year that they're paying you for interest, but at the same time they have to be able to give you back your 1 billion at any time you ask for it.
Lastly, putting that money in bonds, index funds, and other such things scales well. The wealthfront example I used took on a billion in deposits in a short timeframe: https://www.cnbc.com/2019/04/23/wealthfronts-new-high-yield-...
To your first point, if you have another entity do that type of investments for you and present it as a bank account with a variable interest rate; then you can claim you have a 1:1 cash backing truthfully, even if the reality is different. If you just stored the money in a normal bank, the reality would stay the same.
Large banks deal with deposits in total orders of magnitude greater than that (the top banks have over $1 trillion each in deposits.) $1bn in a single demand deposit account is stupid for the depositor, but it doesn't move the needle on a major banks total of deposit accounts, and they can deal with it just fine.