Where to stay in Rome?

I normally stay near Termini station, as it's a convenient area for transportation. There are some good restaurants in that area, and it's not difficult to get to other parts of Rome by Metro or Bus.

I'd suggest having a look at the RS Italy guidebook as there are lots of good ideas there for accommodations in different parts of the city.

Centro Storico is best for centrality to sights and walking around ambiance, but it is also where the most tourists are. With that comes the highest concentration of touristy restaurants, like you described. You can avoid them even when in this area by doing research ahead of time or just walking a bit away from the sights and following your instincts. Also, there are exceptions - Armando Al Pantheon is RIGHT near the Pantheon but is wonderful.

There is no uber in Rome (as of last summer, it may have changed, I suppose). Taxis are easy to come by, and I suggest having data and google maps on your phone for bus directions. The bus system is confusing, but google is pretty good at telling you where a stop is and when (approximately!) a bus is coming. Much cheaper than taxis all the time.

Testaccio has great food, but so do other areas (there are a lot of highly recommended restaurants in Prati, near the Vatican, for example). So regardless of where you choose to stay you will be heading to other neighborhoods for sights and food. Choose where feels the most interesting to you, and get comfortable with the bus or have the budget for frequent taxis. A lot of places are also not that far a walk, plug it into Google Maps, be willing to walk 2 miles and you save money AND might pass some interesting hole in the wall places on the way.

I must not have been in the wrong areas of Rome, but I have never seen restaurants "where they try to pull you in" like the ones in Bruges or Brussels or London.

I've had not perfect food sometimes, but never been accosted at the door as I passed.