If Lightning and off-chain transactions are not that secure, what's the advantage of Lightning over just converting BTC to fiat for transactions?
LN transactions are secure and when you are using LN you are still using bitcoin which is decentralized, cheap and secure. It also doesn't have the limitations of fiat such as the need for a middle man (eg banks) to transfer your money and since cost is based on byte size not amount value the cost is going to be a lot less specially for larger amounts. You also have a much higher privacy compared to using centralized fiat.