Both points accurate.
From an elk-only standpoint the effect of any change will be largely dependent on what happens on the private. It’s one thing to reduce pressure on public land, but there is undoubtedly going to be something that pressures elk off public—hunters, weather, forage, elk just being elk, etc. Even if you made cow elk hunting only valid on private land, you still have people pursuing bulls in public.
Similarly, it will take pressure on private to move elk back to public, and that depends on how many elk a given landowner wants (or doesn’t) on their land. There’s always variation and “it depends,” but for an animal prepping itself for winter and trying not to move unless it has to, it has to be given a reason to move. Easing pressure off an area elk have already left might not be enough; they need that reason to go.
Nowadays even landowners that want elk killed still take steps to keep elk on their lands during the season to better facilitate harvest and pressure how they see fit—taking hunters out themselves, using suppressors so the elk stay on their lands to be managed, etc. The notion of pushing elk around and across boundaries to facilitate better management, while it should still be effective, is only effective if access is consistent across a broad area of landownerships, and is getting less and less used with the vast diversity of landowners and management styles we see more often today.