Supply roads south of Rome and American scouts in Rome
Folklore in Rome says that American scouts reached the outskirts of Rome without encountering any opposition just after the landings. However I could not find any documents on that matter.
Probably reaching Rome was perfectly possible, holding it would be a serious headache. Would the status of Open City be respected? To respect it the allies would have to block routes to Rome north (Civitavecchia, Cassia Flaminia and Salaria much more inland) and probably east. Otherwise the choice would be fighting in the streets and breaking the open city status, or, have the odd situation of Allied and Nazi troops wandering in an out Rome without fighting.
The failed objective at Anzio was not to reach and hold the Alban hills (quite gentle) or the hills south of them (rough). To say the truth the allies were understrengh for this objective but with initial boldness and intelligent defense and subsequent reinforcements it could be done. Reaching them would block the Pontina and other coastal roads, the Appia (Alban Hills, southwest) and possibly the Tuscolana (alban hills norteast).
And mostly expose the Prenestina and Casilina and other roads in the inland valley between the Alban Hills and the Appenines. The German divisions defending the thyrennian coast south would have the supplies, if not totaly blocked, very severely hampered. Without blocking this route the german managed to supply them easily. Instead in the Anzio landing the Allies managed to reach the Pontina but not even holding it. In effect all the allies managed is to hamper, but not block, the coastal roads. Roads that were anyway already exposed to the allied navy.
Motorways (orange red) and the Rome loop were not present at the WW2 time. Other roads are older with many back to roman times.