I've heard that many times but I'm not so sure. For starters, if the fees are high, I don't think it would be convenient for unbanked people to make day-to-day payments with Bitcoin, besides they won't be able to do it in many places and the LN is impractical if not relying on third party wallets. The unbanked do not have enough to save much, they live from day to day and for little savings I think Bitcoin is the same as a bit of gold.
That's a naive thinking, if most people leave their money in their home, the criminals rate will increase really high since they know where you leave your money. Using this reason and saying become your own bank with Bitcoin isn't make sense, you're leaving your private key and your hardware wallet in your house too. Even the criminals didn't find it, they can just pointing their gun in your head and you will tell it honestly since you think your life is more important than money.
But we are talking about unbanked. How much savings can the average unbanked have? $1K? I don't even think so, depending on the country. With amounts like that, I don't think there is much risk of being robbed and if someone does it because they know they have savings, it doesn't matter if they have Bitcoin, gold or cash, the most important thing is that no one knows about it.