Traditional Chinese phonology is organized according to initials (声母), finals (韵母), and tones (声调). This is a satisfactory phonological description of Chinese languages/dialects; it's not that different from a more granular description which broke apart finals into glides, vowels, and codas. I can only speculate as to why they decided to do it this way--perhaps that the scholars were describing the categorical distinctions of a dialect group led them to describe sounds in terms of representative characters for contrastive categories rather than descriptions of the specific phonetic realization. In any event, that's how they did it. The fanqie system for indicating pronunciation is a great example of describing phonetics in terms of characters representative of categories.
Many modern texts on 方言 are directed at linguistically inclined people, and they rely on the background knowledge of that general classification scheme. Also, since most modern 方言 descend from Middle Chinese, using the Guangyun system is pretty convenient at unambiguously talking about the phonology of a given 方言. This is especially common with tones. I have read a whole lot of charts like this.
This background knowledge goes beyond just "linguistically inclined people". I know some school children learn pinyin initial-by-initial and then final-by-final. The 注音 system used mainly in Taiwan (also known as Bopomofo, after the first few initials) is organized into initials and finals, though they also break out the -u- and -i- medials (e.g., huang = (h)(u)(ang)).
Now, to your original question. Why in the hell would a dictionary be organized this way? I can't say I entirely know. A few thoughts:
- There isn't a standard romanization for Sichuanese that is taught in schools, the way standard pinyin is for MSM.
- 方言 books (that don't suck) tend to be written by linguists and use traditional linguistic terminology.
- Chinese books are usually aimed at Chinese people, after all, so they tend to use Chinese characters rather than romanizations. I will say that I sure as hell prefer a traditional-Chinese-phonology approach over a "I haphazardly attempt to transcribe a language using Mandarin" one.