It is true that Austen doesn't rely on symbolism the way--say--Hawthorne does! Nonetheless, there is some symbolism in her works. In Persuasion, the setting is highly symbolic: Anne is facing the autumn of her romantic life--if not her chronological life. In Sense and Sensibility, the mention of Hamlet has symbolic significance: the death of a father and a romance that drives a young woman to distraction. In Emma, Emma's painting of Harriet has symbolic value: Emma is creating Harriet in a new image. In Pride and Prejudice, it may be argued that locations have symbolic importance. Rosings is the symbol of blind-sighted authority. Pemberley is the symbol of gracious authority and true inner virtue. Bath--well Bath is symbolic of the temptations that Lydia embraced. The Lake District symbolizes the poetic ultimate of happiness. Derbyshire symbolizes real, actual happiness. Longbourn (i.e., long-borne) symbolizes the long-borne irritation given by a silly and uncultivated, uneducated woman (no offence intended to feminism). There are undoubtedly others in each, including Pride and Prejudice, so while it is correct to say there is scant use of symbolism, it is not true to say there is no symbolism, even in Pride and Prejudice.
I don't know if there is a single reason for this observation about Austen's novel, but it is an interesting point. My suggestion as to why this is the case would be to look at the development of the novel as a genre and consider Austen's place in that development. Austen wrote her novels at the very turn of the 19th century -- over 200 years ago, and at a time when the novel was still a young form of literature. The earliest novels were "novels of letters" where the "plot line" of the story was told almost exclusively through letters that characters wrote to each other. It was almost like reading a series of short stories in personal narrative form. You can see the remnants of this style in Austen's novel -- she shows over 20 letters between characters. In her more modern development of the novel her characters are all living in the same place and having day-to-day interactions, but the focus is very much on the PEOPLE and their conversations and social gatherings. Austen is a master of characterization and conversation. She doesn't use symbolism as a method of characterization. Symbolism isn't really a feature of this time period -- it becomes more prevalent in the later Romantic novel writing, such as Mary Shelley.