After 45 views and no answer I guess there is no easy solution…
May I ask if by “screen saving” you mean just “screen lock” as in “screen goes black for privacy while I’m occasionally not at the keyboard” or you mean “the computer goes to sleep state to preserve power”?
If the latter, there are ongoing problems since ages with Nvidia GPUs not resuming from sleep in some configurations, but screen locking should work, since the GPU is active.
On a side note, your RTX4070ti supports the 595 series driver (aka G07 driver) so you may consider upgrading to that, maybe not just right now since apparently there are some issues with the very last version, please see black-screen-on-nvidia-after-updating-to-20260428
Also, X11 is barely maintained these days, so time to at least check Wayland and if possible switch to it.