If it's downloadable, it's addable. If it's not downloadable, it's not addable.
If it's public, other people can see it, if it's private, other people can't see it. There used to be a feature called hidden but 3dvia seems to have gotten rid of it for models...
Not even the Send To button can solve the problem. It has to be downloadable to add, and Public for other people to see.
If you have the attribution license, you are assuming that everyone has complete access to it. Therefore, it is always downloadable with attribution license. Change the license to the one that fits your needs. The license has absolutely nothing to do with publicity, so you can change the license and always have it either public or private to your liking.
I'm afraid that if you want to add models to your scene they have to be downloadable.