I would say it is a good thing to have a hard copy of the Holy Qur'an as this book alone has blessings in itself.
Why are we not allowed to touch the written letters of the Qur'an when we are not in a state of purity when the actual writing has not a blessing in itself without even reading the verses.
Muslims kiss the Qur'an and place their forehead on the holy book whilst holding it in their hands or they place it on their head or walk under it before setting off to a journey. Why would they do so if they did not believe that the blessings of the Holy book are transfered to them.
Of course the Qur'an was and is widely memorized, but it is the Holy book being preserved in written form. The followers of the school of Ahl-ul-bayt believe that the revelations to the Holy Prophet were written down on the Prophet's command by Ali ibn Abi Talib during the lifetime of the Prophet.
Many other companions were chosen by the Prophet to write down word for word under his direct supervision.
During the lifetime of the Prophet, many of the companions - fourty-one of whom are recorded by Ibn Nadeem (Al-Firist,p.41)- had written the entire Qur'an with their own hands; therefore each copy was known as "the copy of Abdullah b. Masud", "the copy of Ibn Abbas" and so on.
Sayed Moustafa al-Qazwini, When Power and Piety Collide, p.103
So the Prophet made sure that the revelations were written down accurately to avoid any changes- how could it be otherwise when it was vital to preserve the words of Allah to mankind.
The Holy Qur'an itself attests the preserved status before the life of the Prophet ended.
So, whoever so wills may pay heed to it. It is (recorded) in those scripts (of the Preserved Tablet) that are honoured, exalted, purified, in the hands of those scribes who are honorable, righteous. (80:13-15)
And they said, .(These are) the tales of the ancients he (the messenger) has caused to be written, and they are read out to him at morning and evening. (25:5)
A messenger from Allah who recites the purified scrolls... (98:2)
It is recommended to look at the written words of the Qur'an and to read from a hardcopy for the benefit of our eyes.
Even though we are now living in a digital age and hundreds of books can be stored on a tiny chip for easy and convenient access, we cannot compare the Holy book to any other book. We do not need the latest novel or a heavy dictionary as a hardcopy in our bookshelf but as a Muslim, we need the Holy Qur'an in Arabic writing in our bookshelf as it blesses our home.
You are perfectly right when you say "but there seems to be something special about owning an Arabic Qur'an."
Saying this does not mean that I think having a hardcopy at home is sufficient, of course the utmost blessings come from reading, understanding and applying the holy verses.
There are beautiful copies available in Arabic/English, which make it easier to understand the verses whilst trying to read them in Arabic or follow an audio recitation.