I cannot speak for the US, but in the UK the submission of the reference is not too time critical and becomes relevant only after shortlisting.
However, be careful not to nag the prof too often. You seem to know that the prof hasn't submitted it - if you speak to them in person, you can ask them about it. I would expect a reminder a few days before the deadline, and, if not confirmed, on the day of the deadline should be enough.
It's not clear what "constant reminding" means in your case, but it would really depend on the prof - usually they know it is important to write references and will not forget. If they forget, they are unreliable, but I am not sure constant nagging will help there.