Lightning Service Providers are not custodians. LSPs are usually for-profit service providers who make it easy for users to onboard the Lightning Network in exchange for a fee.
LSPs don't need to control your funds to be able to serve you by connecting you to the Lightning Network and enabling you to make instant Bitcoin payments. LSPs don't control your keys and are only capable of closing the channel they've opened, which causes your funds to go back on-chain.
Note that to be served by an LSP, you need a Lightning node. If you didn't have a Lightning node, you wouldn't be able to be served by any LSP because they wouldn't have a node to connect to.
Self-custodial wallets such as Bitkit have Lightning nodes running inside of them, while custodial wallets don't — users of custodial wallets rely on a single or a handful of nodes controlled by the wallet developer. That's the key difference, and that's why LSPs only serve users of self-custodial wallets.
A channel opened by an LSP is like any other Lightning channel, so it retains the same properties of being enforceable on-chain. The difference between a channel opened by an LSP and a channel opened by any other peer is that LSPs manage liquidity professionally and provide an easier onboarding experience. If your node is connected to a peer node that has good liquidity, your payment success rate will increase. LSPs are not hobbyist node operators, so they manage liquidity professionally, which means that whichever node has a connection to them will have better payment reliability.
To learn more about LSPs, read The Rise of Lightning Service Providers.