Self custody wallets do not rely on LSPs to make transactions they rely on the liquidity of your node and specifically your outbound channels.

An LSP can help you out by opening a channel to you thus letting you receive larger payments but your outbound is specifically tied to how much you have tied up in channels going out to other nodes.

Dual-funded channels are changing that so you can open a channel and if both parties fund it bi-directionally then you have both inbound and outbound in one channel.