If you don't want people to feel that this is expected then just tell them that you don't expect immediate responses.
By scheduling the message to send in the morning you are just setting the expectation to be awake and respond then, you aren't removing the issue you mentioned, you are just picking a different time to set the expectation.
By scheduling the message to send in the morning you are just setting the expectation to be awake and respond then, you aren't removing the issue you mentioned, you are just picking a different time to set the expectation.