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).
Yes you can run LN in a non-custodial manner, but the user experience is really bad.