You are missing the key trick - hedging!

The person who sold the put to you can short some shares and lend out some money to create a "replicating portfolio" that inversely mirrors the payoff of the put option they sold you. They're making a small % on selling the put and then immunising themselves from the market movements.

Who lent them the stock to short? Index funds! The people who want to make market returns, still do. Everyone is happy

Typically, the other party is required to hold a margin account. If you sell a put for 100 units at $50, and the spot (current) price is $45, you're ($50-$45)*100 = $500 in the red. Your broker requires you in that situation to hold at least $500 (usually more) in cash (or cash-equivalent securities) in your margin account.

If the price dips even more, your broker will ask you to add more money to your margin account quickly, or they will liquidate your position. That means that when you don't keep enough cash to cover the loss, your broker will use the cash in your account to buy a put at $50, effectively zeroing out your position. You lose the money you had in margin account, but you are no longer losing (or gaining) money on market movements.

Traders selling options need either cash or margin to cover the trade (according to various calculations, not always the whole amount).

Their broker also monitors the risk and will liquidate any position if they feel it's too risky, and they have their own funds and insurance to cover trades.

Also the options exchange itself has funds and insurance to guarantee the contracts. There's enough money and liquidity in the overall market that it's extremely unlikely for you to worry about this counterparty risk.

It's more of a concern if you're a major investment fund making 9 figure moves and want to make sure banks remain solvent.

That's thinking really far, good stuff. The trading company (Ameritrade for example) insures up to a certain point. Ameritrade wouldn't let you make that trade if you don't have enough capital to back that trade. Lets say you somehow managed to do that trade, and you declare bankruptcy. I think then Ameritrade would have to cover that cost. Then if they can't, then Ameritrade declares bankruptcy. That's as far as I reasoned.

Put options writers need to post cash collateral to cover their obligations or go on equivalent margin (which will be monitored and enforced by their broker) in order to sell you a put.

If this fails, there's still Options Clearing Corporation managing $120B+ collateral and acting as ultimate guarantor for option contracts.