I need to notify systemd that my service has started up successfully, and a task it needs to run after startup requires that the server is already listening
You may use Microsoft.AspNetCore.Hosting.IApplicationLifetime:
///
/// Triggered when the application host has fully started and is about to wait
/// for a graceful shutdown.
///
CancellationToken ApplicationStarted { get; }
Look into this SO post for the configuration example.