Are they being the first country for the sake of being the first country?
Its the same in Europe. Russian Oligarchs bought British Soccer Clubs so that they could become public personalities and avoid being assassinated by the Russian Government or other oligarchs.
The political class in South America exists in a different sphere. They are ancestrally directly European, whereas the working classes of the country are ancestrally native American or mestizo. The concern for the every day worker is minimal as is the overlap between the worlds of these groups.
But smart move. Even the poorest of the poor have smartphones with a (limited) internet subscription.
So a web based wallet like www.coins.io will work wonderfully for them.
But the reason El Salvador is working together with Strike.me is because they have a service where someone pays you in bitcoin but you can choose to receive all or part of the payment in dollars or another currency.
Edit: Lightning networks are bandaids on a prototype cryptocurrency.
I don't know the implementation details of the strike app, but I don't see why you can't keep the signing keys on the device itself (as opposed to the server), and the server is only used for communication with the bitcoin network. That way you don't need to run a lighting node, but they don't control your funds either. Thin wallets use this model I believe.
Yes you can run LN in a non-custodial manner, but the user experience is really bad.
You seem to be correct, based on their suggestion that you download another lightning wallet[1]. My prior impression was that strike contained a lightning wallet and a custodial wallet (for their fiat exchange). That said, with low cost and instant lightning network transactions you can basically use it in a non custodial way (ie. transfer out the moment you exchange for fiat).