Why do we sometimes say that we can use momentum conservation if time period is small and other we don't?

"The conclusion was that you can do it if time period of collision is small". Here you blatantly ignore other assumptions regarding momentum conservation of colliding bodies which though not mentioned in your previous discussion, still need to hold true for momentum conservation. That assumption is that the concerned force should be "non-impulsive" and if it is "impulsive" it should be internal to our chosen system of particles. Here "impulsive" is a relative term and refers to sudden forces which change momentum appreciably even if collision times are negligible because thier larger magnitude of force makes the product F.dt i.e. impulse, considerable. Collision forces are in general impulsive, like the normal force a body exerts on other. Imagine dropping a ball to ground. Though its collision with ground is very fleeting, its momentum changes, infact reverses. This is beacuse the normal acting on ball from ground is an impulsive force. When considering two colliding bodies the normal reaction is still impulsive but as it acts equally and oppositely on both it becomes an internal force. The whole point of making time period small was to prevent the " non impulsive forces " from contributing to momentum change. So as long as there are no external impulsive forces on our chosen system the quoted statement holds true.

Now for the given scenario the normal force from incline is impulsive disallowing momentum conservation.