Wallet of Satoshi doesn't do swaps, it's a fully custodial wallet. Fees for lightning are unaffected by high on chain fees, but your not actually in control of your sats, they are. Best to be used for small amounts only, then move to a self custody wallet once the amount is high enough.

Muun uses swaps, and is self custody. Aqua wallet is another option that uses swaps as well, but on the liquid L2 network, and therefore has lower fees.