Is it the case that the President of the United States will...
Is it the case that the President of the United States will not be able to perform his job and carry out his responsibilities except if he opened the door of the White House for me so that I can go to him whenever I want, talk to him about whatever I want, to waste his time however I want, and to contact him via phone whenever I want? If he does and allows that, will that make him a good President?
And if he doesn't, will he be considered incapable of performing his duties simply because I am not able to relate and deal with him, and just because I cannot contact and communicate with him the same way that I do with my neighbor, friend, or son? Of course not!! The nature of your relationship and communication with the Prophets and Imams (AS) should differ from others, not because they're purified or not, nor because they're protected or not.
Rather, since they represent Allah (SWT), with all His divine sacredness, highness, and greatness. You yourself will respect and give great regard and veneration to the Sheikh of your Masjid or any ‘Alim (scholar), and you will communicate with them in a way that you don't with others. Why?! It is because they represent and signify the religion and Allah (SWT) in your eyes. Then imagine how you should relate and communicate with the holy Prophets, Messengers, and Imams!!
It is to be noted that the presence of their Taharah and ‘Ismah has nothing at all to do with this matter. For example, you give great respect and regard to the police messenger of the Chief of Police to you due to what he represents and whom he represents to you. It does not really matter to you whether this policeman has a good record or whether he is efficient in his work or whether he is close to the Chief of Police or not, or whether his clothes are tidy or not.
Regardless, you will leave a distance between you and him because of what he represents and not for any other reason. As for the argument that you will be able to maintain and keep a relationship with the companions of the Prophet (SA) while you will not be able to keep that same relationship with the Imams of Ahlul-Bayt (AS) and that is due to their Taharah and ‘Ismah, this is absolutely true and is even desired and recommended!
Again, the reason here is not because of their purification and protection. Rather, it is because of their roles and what they represent.