Well, since others have given their views on Rome, here’s mine: OP, my personal opinion is that Rome is no more gritty than most other major cities. To be fair, the area around the Colosseum/Roman Forum/Pantheon is not sanitized in the way that the area around, say, Times Square is, but this is something I actually kind of appreciate about Rome. Rome, unlike other parts of Italy like Venice, the Cinque Terre, and central Florence, is not forced to almost completely cater to the demands and fears of visitors, and that is also something I appreciate greatly about the city.
Unlike the places I named in the previous paragraph–two of which I adore, by the way, but for very different reasons than I appreciate Rome–I see Rome as a real city, whose primary concern is for its many residents who are living, working, studying, and raising families. As the seat of government in a country that loves the art of the protest, there are moments of disruption and violence that look somewhat scary to American citizens, who have little stomach for these things (don’t worry–neither the Italian authorities nor the protestors have any desire for you to become involved and these things are carefully organized).
There is so much to do in Rome that it is overwhelming; this is what gave me anxiety about the city at first. If you visit Rome you will have the opportunity to expose yourself to many important Classical, early Christian, late Renaissance, and Baroque works of art and architecture (to name the highlights only). If you’re a 20th century history person and willing to do some basic research online, you can learn a lot about neighborhoods and urban projects that developed in support of, and then as a reaction to, Fascism in Italy. With a little bit of research and care, you can find places that serve lovely versions of traditional dishes like carbonara or coda, and, unlike many other common tourist cities in Italy, you can also find fine non-Italian restaurants in the city. You can take the trams out to the outlying neighborhoods and simply watch people. You can go to one of the big open spaces like the Villa Borghese or the Villa Pamphilij and picnic, or walk along the Passeggiata del Gianicolo and get a great view of the city.
I would absolutely agree that any visit to Rome that just focused on the “expected” stuff (the Colosseum, the Trevi, the Spanish Steps, the Borghese Gallery, the Vatican, etc.) would lack something. The Trevi’s just a fountain and does not, in and of itself, have special powers, La Dolce Vita aside. The romance and magic come from the spontaneous and felicitous combination of the beauty of the setting, the plesant company of one’s companion, and a certain (pardon the French) je ne sais quoi. When the Trevi is scaffolded or surrounded by a million people who are all desperate to have the same special WOW in the fifteen minutes that they gave themselves to be at the fountain before they have to move on to the next thing on the list, of course it will disappoint. So–find another fountain, find another statue, find another quaint street. The city is littered with them, and with the opportunities to have that magical feeling, if you’re willing to push things just a little!