Here is how it worked in Northern Cameroon, without refrigeration (which was too expensive and considered fairly unnecessary.).
General meat consumption was low. Unless you are filthy rich, eating meat (besides dried fish) at home is going to be something you do once a week or so. Most meals are vegetarian out of hardship. All meals are generally prepared from stored staple grains and small bits of food bought from the market or neighbors on a day to day basis. Nobody buys a week, or even a couple days, worth of groceries at a time. Even if you wanted to, it’d be way to expensive for most people, who live a penny by penny existence People even by single cloves of garlic and tiny single-serving bags of spices. That’s usually all the money they have that day will allow.
If you are eating chicken or another fowl, you are going to go find a live one (either from the market or your own stock), prepare it and eat it. Chickens are pretty small (especially without the hormones and feed we are used to) and meat is rare, so leftovers are unlikely. If for some crazy reasons there are leftovers, it’d be boiled for hours in the next day’s stew.
Fish is almost always salted and dried, either at home or on a small scale commercial level. The dried fish is added to stew whole, and this is an average Joe’s main form of animal protein. Fresh fish is eaten near rivers and coastlines, and these days frozen fish is distributed to vendors, but in general if you aren’t near a body of water, dried fish is where it is. People generally buy just enough for that night’s meal, and leftovers are unlikely.
Goats and sheep are a big deal. If you kill a goat, you are having a party. You aren’t eating it all yourself. The big time that this happens i Tabaski, when every Muslim family slaughters a sheep (or goat.) For one or two days, everyone stuff themselves with meat, and gives piles of meat to everyone. You are expected to share with the homeless and those too poor for their own animals. A little bit of meat will be preserved for later snacking by deep frying it and storing the charred cubes in oil, but for the most part if you have leftovers, it’s considered a little shameful. You are expected to share. But, if for some reason you did have some extra meat, you’d put it in stew, which would be boiled for hours the next day.
Cows aren’t really prepared for home use. In small communities, a cow may be slaughtered for festive occasions and the entire village will eat it. In larger communities, one or more cows may be killed a day and sold at the market. This meat is generally bought first thing in the AM, prepared for the noon meal, and eaten again as leftovers at dinner. Nobody buys more than they can eat in a day. There are not usually leftovers. If you are shopping for meat at, say, 1 PM. you probably aren’t going to find any left. The meat sellers will have already sold that day’s stock and gone on to their other job.
In summary, meat consumption is low, and eating meat that wasn’t freshly slaughtered would be rare. I am fairly certain that Northern Cameroon’s patterns are relatively consistent among other fairly poor, warm countries. The way that temperate climates manage meat is extremely different, and out there the “spices cover rotten meat” theory might make some sense. It might especially be true in the urban, buyer beware culture of the early age of exploration. But it’s not consistent with the way food is consumed elsewhere.