From what I've read/heard, this is a popular misconception, with no basis in reality. I'm going off of the views of Massimo Montanari (unfortunately I only know his Italian sources: e.g., Gusti del Medioevo: I prodotti, la cucina, la tavola and a series he did for RAI2, La Storia in Cucina ); I'm not a food historian, though, so hopefully others with more expertise can also weigh in.
Montanari's basic argument on this topic is that spices were primarily (perhaps exclusively) the domain of the very wealthy and powerful—i.e., the type of people who would not be eating rotting meat. Rather the people who bought spices could afford the very best meat, and the freshest. Spices were simply another way of displaying one's wealth and ability to bring in goods from distant places. In short, another mode of conspicuous consumption (literally).
Also, while people in lower rungs of society might eat less (or no) meat, or meat of perceived lower quality (such as older animals with tougher muscles), there is nothing I've come across in Montanari to suggest they were eating rotting meat. But again, I defer to food historians here.
Hi, there's a discussion on the spice trade happening right now on the front page. See this thread featuring /u/mikedash
Jacques Barzun has an aside about this in From Dawn To Decadence where he says it’s ridiculous—Europe had plenty of delicious foods/flavorings to use.