They didn't create the society you live in, millennia of history did that, and even in western Europe much of the input was from the peoples who put an end to the Empire.

Do you have an Emperor? No. Do you have Roman law? Possibly some, but not if you're in the English-speaking world. Do you have slaves? Unless you're some radical levelling subversive I'd consider that a must, assuming you enjoy several times an average ancient income. Do you speak Latin? Well yes, as it happens, but as you point out there's usually not much occasion to use it. :)

I have to confess I've never quite fathomed the western attachment to Rome and Greece before it. Rome in particular has left its mark, but my experience is of a world conditioned by later experience, above all of a revolution in material life beyond Rome's wildest dreams.

There's an interesting discussion to be had though about whether Europe owes more to the Classical or medieval epoch. I'm firmly on the side of our more recent ancestors with their restlessness and wonky contractuality against arbitrary Imperial power, slavery and gory spectacle whose provision might merely be my lot.