It was essentially because Admiral was seen as having too many connotations of royalty and nobility, which we can say is a bit hypocritical when we see General in use in the Army.

To make up for it Commodore was indeed used when ships were in squadrons, though it was never an official rank. One of the bridges in Philadelphia is actually called the Commodore Barry Bridge after John Barry who was the first Captain in the new USN and its first senior officer. And for a time the special rank of "Flag Officer" was created. However it did cause some sting for USN Captains who could never be official equals to their European peers, though often the RN especially would still treat senior captains and squadron commanders with appropriate honors.

For much of the time even during war the fleet was so dispersed as to rarely have more than a few ships in company, and even then few of them were large enough to rate a full Captain to be in command. The Navy rarely had more than a few dozen frigates in service at a time and for all intents no ships of the line meaning the number of true Captains at sea was small and generally on station alone for lengths of time. Not really an organization in need of many Admirals to begin with.

From the USN itself: https://www.history.navy.mil/research/library/online-reading-room/title-list-alphabetically/w/naval-traditions-names-of-rank/officer/admiral.html

While Ian W. Toll's 6 Frigates is another great look at the USN through the end of the war of 1812.