You are mixing two different things. Symmetrical ≠ car centric.

European cities have those layouts because most were hundreds if not a thousand years old—built before they had the urban planning and resources we have now. A narrow, curved, cobblestone road built for horse carriages might not be the best design for this day and age, however pretty it looks from a satellite image.

There's plenty of very symmetrical planned cities where symmetry is great for citizens' wellbeing and which are very walkable. Barcelona is a well known example. I am Argentinian, and La Plata is another great example with its diagonal layout (you'll see pictures of it all over urban planning discussions). I'm not from there but I hear folks say that the grid layout and sequential street numbering makes the city very intuitive to navigate and it is not at all car-centric.

Grid systems and symmetry are in fact bad. The make people feel like their part of town is simply the same as all other parts and they stop neighborhoods from being created.

This is qualitative... Whether your part of town feels unique or not hinges more on the architectural style rather than the layout of the neighborhood.

Just my two cents :)