Has it truly been established that the restaurant industry operates on thin margins?
TLDR: Interesting question, but difficult to answer. The restaurant industry is made up of so many firms and many are small and self-owned. This makes collecting data difficult. The data I did collect shows restaurants operate with margins that are tight, but not as tight as commonly believed.
A potential hypothesis of the reason behind this is restaurants are often small enterprises run by a family. Those businesses may operate on healthy margins in good times but struggle excessively in bad times. Spending at restaurants is highly variable and falls dramatically during recessions. This triggers a wave of failures across smaller restaurants who do operate on tight margins and are unable to secure financing thus causing them to fail. This occurrence might be a large contributing factor to the commonly held notion that restaurants operate on extremely thin margins.

Forbes wrote a piece on restaurant margins in early 2018. The data they complied showed margins to be 6.5% for the most recent year. That is low but certainly not razor thin. However, 2018 was deep into an economic expansion that restaurants were certainly benefiting from. The numbers from earlier years is much tighter. In 2014 it was 2.3%, 2.3% again in 2015, 2.8% in 2016, and 3.3% in 2017. Those numbers paint a different picture which is much gloomier for restaurants.


The dataset posted in the comments does not contain information on profit margins but it does contain information on the proxy measurements return on equity and return on assets. The dataset was published February 26, 2020 and contains a selection of 861 restaurants from across Europe.
For the United Kingdom the average return on equity for a restaurant is 18% and the average return on assets is 4.47%. Spain has an average RoE of 19.3% and RoA of 7.3%. France has an average RoE of 4.45% and RoA of 1.9%. Besides France, these are robust numbers.

The Australian Taxation Office collects data on many industries including the restaurant industry. Average total expense for a small restaurant is 85% (65,000-500,000) and for a large restaurant (2,000,000+) 92%.

There is an interesting comparison of margins at McDonalds and high-end restaurants over on Quora.