I think it wasn't so much about the Americans getting there first so much as Mark Clarke.
When breaking out of Anzio the decision to go for Rome instead of cutting off the German army in southern Italy to me was his biggest mistake.
In my opinion many of his decisions seemed to be made for his personal glory.
Mark Clarke's generalship is a very controversial subject.
He does have his defenders though.
This is a good article at HistoryNet.com
Does he really deserve to be exhibit A in the war's pantheon of bad commanders?
www.historynet.com
A piece of that article:
:…….Finally, the main indictment: his decision to drive for Rome after the Anzio breakout rather than encircling the German Tenth Army, which was at that moment retreating north in some confusion.
But this accusation, too, fails the evidence test. After all, surrounding a maneuver-trained German field army was not as easy as it sounds. How many times did the Western Allies ever succeed in doing so? Don’t try too hard, it’s an easy answer. Before the final German collapse in 1945: zero.
So if Clark is culpable for failing to encircle a German army in battle, he has some very fine company: Generals Dwight D. Eisenhower, Omar Bradley, Courtney Hodges, and George S. Patton Jr. It was possible to beat the Germans, yes. Outside of a few extraordinary circumstances in this war, however, they usually maneuvered rapidly enough to prevent themselves from being encircled—and that is exactly what they did in Italy.â€
- HistoryNet article