I'm looking for a general rule to apply to all current collections so that only products where inventory is greater than 3 are published/visible.
I understand a simple collection can be created such as:
1. Inventory is greater than 3
2. Condition x
but this is too limited. Our collections are based on multiple OR (any) conditions
1. title = x
2. vendor = y
3. tag = z
So if we add the condition that
3. inventory > 3
the results will include products where inventory is less than 3.
It's possible that you could turn on/off product visibility with Shopify Flow, by setting up triggers and conditions based on the scenarios you've described. Have you explored using this yet?