Using 反切 to determine readings of characters in modern varieties of Chinese
Sound shift, although an important factor, is not the primary reason that Fanqie doesn't fully work for determining the sound of characters in modern varieties of Chinese. In actuality, Fanqie never fully worked for any variety of Chinese, modern or ancient. This is because the phonology recorded in rime dictionaries wasn't for any particular language, but at the very outset was a compromise between Northern and Southern varieties - that is, Middle Chinese was itself artificial, never being a natural language.

　　昔開皇初，有儀同劉臻等八人，同詣法言門宿。夜永酒闌，論及音韻，以今聲調既自有別，諸家取捨亦復不同。吳楚則時傷輕淺，燕趙則多傷重濁；秦隴則去聲為入，梁益則平聲似去。又支、脂、魚、虞共為一韻；先、仙、尤、侯俱論是切。欲廣文路，自可清濁皆通，若賞知音，即須輕重有異。
　　呂靜《韻集》、夏侯該《韻略》、陽休之《韻略》、周思言《音韻》、李季節《音譜》、杜臺卿《韻略》等各有乖互。江東取韻，與河北復殊，因論南北是非，古今通塞，欲更捃選精切，除削疏緩；蕭顏多所決定。魏著作謂法言曰：「向來論難，疑處悉盡，何不隨口記之。我輩數人，定則定矣。」法言即燭下握筆，略記綱紀，博問英辯，殆得精華。
　　於是更涉餘學，兼從薄宦，十數年閒，不遑修集。今返初服，私訓弟子，凡有文藻，即須明聲韻。屏居山野，交遊阻絕，疑惑之所，質問無從。亡者則生死路殊，空懷可作之歎。存者則貴賤禮隔，以報絕交之旨。遂取諸家音韻、古今字書，以前所記者，定為《切韻》五卷。剖析豪氂，分別黍累，何煩泣玉，未得縣金，藏之名山。昔怪馬遷之言大，持以蓋醬，今歎楊雄之口吃。非是小子專輒，乃述群賢遺意，寧敢施行人世，直欲不出戶庭。于時歲次辛酉，大隋仁壽元年。
—— 陸法言《切韻・序》

Some time ago, during the first year of the 開皇 era, I was host to 劉臻 and seven other ministers (蕭該, 顏之推, 盧思道, 李若, 辛德源, 薛道衡, and 魏彥淵). After an evening of joyous drinking and feasting, the discussion turned to the subject of phonology.
The sounds of previous eras and the other geographical regions are different, and the subject had different schools of discipline: the 吳楚 (southern) topolects were light and gentle; the 燕趙 (northern) topolects were heavy and thick; the 秦隴 topolects (now around Shaanxi and Gansu) read the departing tone as the checked tone; the 梁益 topolects (now around Sichuan) read the level tone as the departing tone. Some schools treat「支」and「脂」,「魚」and「虞」with the same rhyme; some treat「先」and「仙」,「尤」and「侯」with the same initial. If one wished to indulge in a wide variety of literature, then one can personally interchange light and heavy sounds; however, if one wished to understand phonology, then one must clearly differentiate them.
The subject matter given in 呂靜《韻集》, 夏侯該《韻略》, 陽休之《韻略》, 周思言《音韻》, 李季節《音譜》, 杜臺卿《韻略》 and others were all different to the point of being contradictory. The phonology of 江東 (Jiangnan) is completely different to that of 河北 (northern regions). Thus, we discussed differences between the northern and southern topolects and between past and present, leading on to the choice of a precise and refined phonology standard, freed from clumsy and inelegant representations of sound, with many of the choices made by 蕭該 and 顏之推. 魏彥淵 later wrote to me: &quot;That was a challenging discussion, but the most difficult of problems have been solved. Why don't you record the results of our conference? We'll treat the points that we have agreed upon as the established rules [of phonology].&quot; Holding a brush under the candlelight, I proceeded to summarise the phonological rules decided, the studious questions asked, and their incisive responses of that evening, attempting to capture their quintessence.
Over the next ten to twenty years, the knowledge established that evening underwent ceaseless additions from knowledgeable scholars and officials. Today, I've regressed to my roots, and have taken on students; as long as they compose works, they must understand the rules of phonology. I'm living among the mountains; ceasing to travel on far journeys for academic pursuits, the questions that I harbour have no place to be discussed. As for the old friends and colleagues of mine: those who have long since passed away, I impossibly wish for their resurrection; and those who are still alive have gone down different paths in their lives, and I wish to sever ties with them. From the phonological studies taken from the different schools and literature on the subject matter from the past to present, the body of knowledge amassed is composed into the five volumes of Qieyun. Minute differences are stressed and fine details are emphasised, whilst not being obstinate and conceited [in its opinions]. It is truly an illustrious composition, worthy as a treasured legacy for many generations. I previously criticised Sima Qian for the veneration of his own work, but now I'm fond of teasing those who gaze on, open-mouthed in awe [at such grand creations]. However, I don't dare to presume the work's propensity towards widespread popularity, and only wish  for it to be used within the households.
Preface to Qieyun, by 陸法言
辛酉 year, 仁壽 era of the Sui Dynasty