... did he realize his own limitations as a Don?...
The answer is no, he didn't.
But to be fair to Sonny, he would've had neither the time nor the patience to sit around and ponder whether or not he was a good, bad, or adequate Don in comparison to his father. Thrust into this leadership role quite suddenly, he was doing in his opinion the best job he could, running the Family, avenging the attack on his father, and boinking Lucy any time he had an hour to spare.
Apple